Theory of Bridge Aerodynamics

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Theory of Bridge Aerodynamics

Einar N. Strømmen

Theory of Bridge Aerodynamics

ABC

Professor Dr. Einar N. Strømmen Department of Structural Engineering Norwegian University of Science and Technology 7491 Trondheim, Norway E-mail: [email protected]

Library of Congress Control Number: 2005936355 ISBN-10 3-540-30603-X Springer Berlin Heidelberg New York ISBN-13 978-3-540-30603-0 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and TechBooks using a Springer LATEX macro package Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper

SPIN: 11545637

89/TechBooks

543210

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 August, 2005

Einar N. Strømmen

CONTENTS Preface Notation

1

INTRODUCTION 1.1 1.2 1.3 1.4

2

3

4

5

General considerations Random variables and stochastic processes Basic flow and structural axis definitions Structural design quantities

vii xi

1 1 4 6 10

SOME BASIC STATISTICAL CONCEPTS IN WIND ENGINEERING

13

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10

13 15 27 30 33 38 41 43 44 45

Parent probability distributions, mean value and variance Time domain and ensemble statistics Threshold crossing and peaks Extreme values Auto spectral density Cross spectral density The connection between spectra and covariance Coherence function and normalized co–spectrum The spectral density of derivatives of processes Spatial averaging in structural response calculations

STOCHASTIC DESCRIPTION OF TURBULENT WIND

53

3.1 3.2 3.3

53 58 63

Mean wind velocity Single point statistics of wind turbulence The spatial properties of wind turbulence

BASIC THEORY OF STOCHASTIC DYNAMIC RESPONSE CALCULATIONS

69

4.1 4.2 4.3 4.4

69 76 81 84

Modal analysis and dynamic equilibrium equations Single mode single component response calculations Single mode three component response calculations General multi–mode response calculations

WIND AND MOTION INDUCED LOADS 5.1 5.2 5.3

The buffeting theory Aerodynamic derivatives Vortex shedding

91 91 97 102

CONTENTS

6

7

8

x

WIND INDUCED STATIC AND DYNAMIC RESPONSE CALCULATIONS

109

6.1 6.2 6.3 6.4

109 113 116 142

Introduction The mean value of the response Buffeting response Vortex shedding

DETERMINATION OF CROSS SECTIONAL FORCES

157

7.1 7.2 7.3 7.4

Introduction The mean value The background quasi static part The resonant part

157 163 163 182

MOTION INDUCED INSTABILITIES

195

8.1 8.2 8.3 8.4 8.5

195 199 200 201 203

Introduction Static divergence Galloping Dynamic stability limit in torsion Flutter

Appendix A: A.1 A.2 A.3 A.4

TIME DOMAIN SIMULATIONS

209

Introduction Simulation of single point time series Simulation of spatially non–coherent time series The Cholesky decomposition

209 210 213 221

Appendix B: B.1 B.2

DETERMINATION OF THE JOINT ACCEPTANCE FUNCTION

Closed form solutions Numerical solutions

Appendix C:

References Index

AERODYNAMIC DERIVATIVES FROM SECTION MODEL DECAYS

223 223 223

227

233 235

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 . d iag [⋅] is a diagonal matrix whose content is written within the brackets.

d et ( ⋅) 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 p is short for return period.

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 , σ 2 is the standard deviation, variance.

µ is a quantified small probability. Imaginary quantities:

i is the imaginary unit (i.e. i = −1 ). R e[⋅] 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 , r ) indicates time derivatives, i.e. d / d t , d 2 / d t 2 .

xii

NOTATION

A prime on a variable (e.g. C L′ or φ ′ ) indicates its derivative with respect to a relevant variable (except t), e.g. C L′ = d C L d α and φ ′ = d φ d x . Two primes is then the second derivative (e.g. φ ′′ = d 2φ d x 2 ) and so on. Line ( − ) above a variable (e.g. C D ) indicates its average value. A tilde ( ∼ ) above a symbol (e.g. M ) indicates a modal quantity. i

ˆ ) indicates a normalised quantity. A hat ( ∧ ) above a symbol (e.g. B The use of indexes:

Index x , y or z refers to the corresponding structural axis.

x f , y f or z f refers to the corresponding flow axis. 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. m ax,m in 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.

L exp

∫

L

means integration over the entire length of the structure.

xiii

NOTATION

Latin letters Aerodynamic admittance functions (m = y, z or θ, n = u or w)

Am n

A1*

−

A 6*

Aerodynamic derivatives associated with the motion in torsion

a

Constant or Fourier coefficient

B

Cross sectional width

ˆ Bq or B q

Buffeting dynamic load coefficient matrix

b

Constant, coefficient, band-width parameter

bq or bˆ q

Mean wind load coefficient vector

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 Eˆ , Eˆ

Modulus of elasticity Impedance, impedance matrix

e

Eccentricity, distance between shear centre and cetroid

F,F

Force vector, force at (beam) element level

f , fi

Frequency [Hz], eigen–frequency associated with mode i

f ( ⋅)

Function of variable within brackets

G

Modulus of elasticity in shear

G F or GF Influence function or matrix ( F = V y , V z , M x , M y or M z ) g ( ⋅)

H 1*

−

Function of variable within brackets H 6*

Aerodynamic derivatives associated with the across-wind motion

H or H

Frequency response function or matrix

Ip

Centroidal polar moment of inertia

It , Iw

St Venant torsion and warping constants

Iu , Iv , Iw

Turbulence intensity of flow components u, v or w

Iy , Iz

Moment of inertia with respect to y or z axis

I

Identity matrix

Iv

Turbulence matrix ( Iv = d iag [ I u

i

The imaginary unit (i.e. i = −1 ) or index variable

I w ] or Iv = d iag [ I u

Iv

Iw ] )

xiv

NOTATION

J

Joint acceptance function

j

Index variable

K ,K

Stiffness, stiffness matrix

k

Index variable, node or sample number

kp

Peak factor

kT

Terrain roughness coefficient

L , L exp

Length, wind exposed length

m

Integral length scales (m = y, z or θ, n = u,v or w)

Ln

Mm ,Mu

Bending moment (m=x, y, z), ultimate bending moment strength

m

Index variable

m or M

Mass or mass matrix

m i

Modally equivalent and evenly distributed mass

N

Number, number of nodes or number of elements in series

n P1*

Index variable

−

P6*

Aerodynamic derivatives associated with the along-wind motion

p

Index variable, node or sample number

Q or Q

Wind load or wind load vector at system level

q or q

Wind load or wind load vector at cross sectional level

qU , qV

Velocity pressure, i.e. qU = ρU 2 / 2 , qV = ρV 2 / 2

R

Load vector at system level

Re

Reynolds number

Rp

Return period.

r or r

Cross sectional displacement or rotation, displacement vector

St

Strouhal number

S or S

Auto or cross spectral density, cross-spectral density matrix

S x (ω )

Single side auto-spectral density of variable x

S x ( ±ω )

Double side auto-spectral density of variable x

S xy

Cross-spectral density between components x and y

t ,T

Time, total length of time series

Tn

Turbulence time scales (n = u,v or w)

U

Instantaneous wind velocity in the main flow direction

u

Fluctuating along-wind horizontal velocity component

V , VR

Mean wind velocity, resonance mean wind velocity

v

Fluctuating across wind horizontal velocity component

NOTATION

xv

v

Wind velocity vector containing fluctuating components

w

Fluctuating across wind vertical velocity component

X ,Y , x , y

Arbitrary variables, e.g. functions of t

x , y, z

Cartesian structural cross sectional main neutral axis (with origo in the shear centre, x in span-wise direction and z vertical)

x f , yf , z f

Cartesian flow axis ( x f in main flow direction and z f vertical)

xr

Chosen span-wise position for response calculation

z0

Terrain roughness length

z m in

Minimum height for the use of a logarithmic wind profile

Greek letters

α

Coefficient, angle of incidence

β

Constant, coefficient

β

Matrix containing mode shape derivatives

γ

Coefficient, safety coefficient

ε0

Mean wind velocity band width parameter

ζ or ζ

Damping ratio or damping ratio matrix

η or η

Generalised coordinate or vector containing N m od η components

θ

Index indicating cross sectional rotation (about shear centre)

κ

Constant, statistic variable

κ ae

Matrix containing aerodynamic modal stiffness contributions

ν

Kinematic viscosity of air

λ

Non–dimensional coherence length scale of vortices

µ

A quantified small probability.

µn

Spectral moment

µae

Matrix containing aerodynamic modal mass contributions

ρ

Coefficient or density (e.g. of air)

ρ ( ⋅)

Covariance (or correlation) coefficient of content within brackets

ρ

Cross covariance or correlation matrix between a set of variables

σ ,σ τ

2

Standard deviation, variance Time shift (or lag)

xvi

NOTATION

Φ

3 ⋅ N m od by N m od matrix containing all mode shapes φi

Φr

3 by N m od matrix containing the content of Φ at x = x r

φ

3 by 1 mode shape vector containing components φ y , φz , φθ

φ yi ,φzi ,φθi

Mode shape components in y , z and θ directions associated with mode shape i (continuous functions of x or N by 1 vectors)

ϕxy

Phase spectrum between components x and y

ψ

Phase angle

ψ ( ⋅)

Function of the variable within the brackets

ω

Circular frequency (rad/s)

ωi

Still air eigen-frequency associated with mode shape i

ωi (V

)

Resonance frequency assoc. with mode i at mean wind velocity V

Symbols with both Latin and Greek letters:

∆f , ∆ω

Frequency segment

∆t

Time step

∆s

Separation (s = x, y or z)

∆x

Span–wise integration step

Chapter 1 INTRODUCTION

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

qU (t ) =

2 1 ρ U (t )  2 

(1.1)

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

2

1 INTRODUCTION

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. Surely, this is only for convenience as there are really no regions where they exclusively occur alone. The important question is to what extent they are adequately included in the mathematical description of the loading process. In structural engineering the wind induced fluctuating forces and corresponding response quantities are usually assumed stationary, and thus, response calculations may be split into a time invariant and a fluctuating part (static and dynamic response). An illustration of what can be expected is shown in Fig. 1.1. For a mathematical description of the process from a fluctuating wind field to a corresponding load that causes a fluctuating load effect (e.g. displacements or cross sectional stress resultants) a solution strategy in time domain is possible but demanding. The reason for this is that the wind field is a complex process that is randomly distributed in time and space. A far more convenient mathematical model may be established in frequency domain. This requires the establishment of a frequency domain description of the wind field as well as the structural properties, and it involves the establishment of frequency domain transfer functions, one from the wind field velocity pressure distribution to the corresponding load, and one from load to structural response. We shall see that this implies the perception of wind as a stochastic process, and a structural response calculation based on its modal frequency-response-properties. The important input parameters to this solution strategy are the statistical properties of the wind field in time and space, and the eigen-modes and corresponding eigen-frequencies of the structural system in question. The outcome is the statistical characteristics of the structural response.

4

1 INTRODUCTION

not be treated in this book. It is taken for granted that the modal damping ratio is known from elsewhere (e.g. standards or handbooks).

1.2

Random variables and stochastic processes

A physical process is called a stochastic process if its numerical outcome at any time or position in space is random and can only be predicted with a certain probability. A data set of observations of a stochastic process can only be regarded as one particular set of realisations of the process, none of which can with certainty be repeated even if the conditions are seemingly the same. In fact, the observed numerical outcome of all physical processes is more or less random. The outcome of a process is only deterministic in so far as it represents a mathematical simulation whose input parameters have all been predetermined and remain unchanged. The physical characteristics of a stochastic process are described by its statistical properties. If it is the cause of another process, this will also be a stochastic process. I.e. if a physical event may mathematically be described by certain laws of nature, a stochastic input will provide a stochastic output. Thus, statistics constitute a mathematical description that provides the necessary parameters for numerical predictions of the random variables that are the cause and effects of physical events. The instantaneous wind velocity pressure (see Eq. 1.1) at a particular time and position in space is such a stochastic process. This implies that an attempt to predict its value at a certain position and time can only be performed in a statistical sense. An observed set of records can not precisely be repeated, but it will follow a certain pattern that may only be mathematically represented by statistics. Since wind in our built environment above ground level is omnipresent, it is necessary to distinguish between short and long term statistics, where the short term random outcome are time domain representatives for the conditions within a certain weather situation, e.g. the period of a low pressure passing, while the long term conditions are ensemble representatives extracted from a large set of individual short term conditions. For a meaningful use in structural engineering it is a requirement that the short term wind statistics are stationary and homogeneous. Thus, it represents a certain time–space–window that is short and small enough to render sufficiently constant statistical properties. The space window is usually no problem, as the weather conditions surrounding most civil engineering structures may be considered homogeneous enough, unless the terrain surrounding the structure has an unusually

1.2 RANDOM VARIABLES AND STOCHASTIC PROCESSES

5

strong influence on the immediate wind environment that cannot be ignored in the calculations of wind load effects. The time window is often set at a period of T = 10 minutes.

Fig. 1.2

Short term stationary random process

Such a typical stochastic process is illustrated in Fig. 1.2. It may for instance be a short term representation of the fluctuating along wind velocity, or the fluctuating structural displacement response at a certain point along its span. As can be seen, it is taken for granted that the process may be split into a constant mean and a stationary fluctuating part. There are two levels of randomness in this process. Firstly, it is random with respect to the instantaneous value within the short term period between 0 and T. I.e., regarding it as a set of successive individual events rather than a continuous function, the process observations are stored by two vectors, one containing time coordinates and another containing the instantaneous recorded values of the process. The stochastic properties of the process may then be revealed by performing statistical investigations to the sample vector of recorded values. For the fluctuating part, it is a general assumption herein that the sample vector of a stochastic process will render a Gaussian probability distribution as illustrated to the right in the figure. This type of investigation is in the following labelled time domain statistics. The second level of randomness pertains to the simple fact that the sample set of observations shown in Fig. 1.2 is only one particular realisation of the process. I.e. there is an infinite number of other possible representatives of the process. Each of these may

6

1 INTRODUCTION

look similar and have nearly the same statistical properties, but they are random in the sense that they are never precisely equal to the one singled out in Fig. 1.2. From each of a particular set of different realisations we may for instance only be interested in the mean value and the maximum value. Collecting a large number of different realisations will render a sample set of these values, and thus, statistics may also be performed on the mean value and the maximum value of the process. This is in the following labelled ensemble statistics. In wind engineering X k = x k + x k (t ) may be a representative of the wind velocity fluctuations in the main flow direction. The time invariant part x k is then the commonly known mean wind velocity, given at a certain reference height (e.g. at 10 m) and increasing with increasing height above the ground, but at this height assumed constant within a certain area covered by the weather system. The fluctuating part x k (t ) represents the turbulence component in the along wind direction. The mean wind velocity is a typical stochastic variable for which long term ensemble statistics are applicable, while the turbulence component is a stochastic variable whose statistical properties are primarily interesting only within a short term time domain window. Likewise, the relevant structural response quantities, such as displacements and cross sectional stress resultants, may be regarded as stochastic processes. In the following, it is to be taken for granted that the calculation of structural response, dynamic or nondynamic, are performed within a time window where the load effects are stationary [i.e. the static (mean) load effects are constant and the dynamic (fluctuating) load effects are Gaussian with a constant standard deviation].

1.3

Basic flow and structural axis definitions

The instantaneous wind velocity vector is described in a Cartesian coordinate system  x f , y f , z f  , where x f is in the direction of the main flow and z f is in the vertical direction as shown in Fig. 1.3.a. Accordingly, the wind velocity vector is divided into three components.

1 INTRODUCTION

8

As mentioned above, the relevant time window is of limited length such that the component in the main flow direction may be split into a time invariant mean value and a fluctuating part. Thus, the instantaneous wind velocity vector is defined by

( ) v ( x f , y f , z f ,t ) w ( x f , y f , z f ,t )

(

)

(

U x f , y f , z f ,t = V x f , y f , z f + u x f , y f , z f ,t

)

     

(1.2)

where V is the mean value in the main flow direction, and u, v and w are the turbulence components whose time domain mean values are zero. Since the main flow direction is assumed perpendicular to the span of the structure, the velocity vector may be greatly simplified depending on structural orientation. Thus, Eq. 1.2 may be reduced to

(

)

( w ( y f ,t )

U y f ,t = V + u y f ,t

)

   

(1.3)

for a line−like horizontal structure (e.g. a bridge), and into

(

)

( ) ( v ( z f ,t )

U z f ,t = V z f + u z f ,t

)

   

(1.4)

for a vertical structure (e.g. a tower). As shown in Fig. 1.3 the structure is described in a Cartesian coordinate system [ x , y , z ] , with origo at the shear centre of the cross section,

x is in the span direction and with y and z parallel to the main neutral structural axis (i.e. the neutral axis with respect to cross sectional bending). Correspondingly, the wind load drag, lift and pitching moment components (per unit length along the span) are all referred to the shear centre and split into a mean and a fluctuating part, i.e. q y ( x )  q y ( x , t )      q + q = q z ( x )  + q z ( x , t )  q ( x )  q ( x , t )   θ   θ 

(1.5)

Similarly, the response displacements ry ( x )  ry ( x , t )      r + r =  rz ( x )  +  rz ( x , t )   r ( x )  r ( x , t )  θ  θ 

(1.6)

1.3 BASIC FLOW AND STRUCTURAL AXIS DEFINITIONS

9

and cross sectional stress resultants

 Q y ( x )   Q y ( x ,t )       Qz ( x )  +  Qz ( x ,t )   M x   M x ,t  ( )  x ( )  x

(1.7)

are also referred to the shear centre, while bending moment and axial stress resultants  M y ( x )  M y ( x ,t )      M z ( x ) +  M z ( x ,t )  N ( x )   N ( x ,t )     

(1.8)

are referred to the centroid of the cross section (where, as shown above, the centriod is defined as the origo of main neutral structural axis).

Fig. 1.4

Structural axes and displacement components

Thus, it is assumed that structural response in general can be predicted as the sum of a mean value and a fluctuating part, as illustrated in Fig. 1.4. It is assumed that within the time window considered the mean values are constant as well as the statistical properties of the fluctuating parts.

1.4 STRUCTURAL DESIGN QUANTITIES

11

Since structural behaviour is assumed linear elastic, these quantities may in general be obtained from the extreme values of the displacements

rk ( x ) + rk ( x , t ) 

max

k = y , z ,θ

where

(1.9)

However, the mean values in this situation are time invariants, and the response calculations have inevitably been based on predetermined values taken from standards or other design specifications. They have been established from authoritative sources to represent the characteristic values within a certain short term weather condition chosen for the special purpose of design safety considerations. Therefore, in a particular design situation time invariant quantities may be considered as deterministic quantities, and thus, the mean values of displacements or stress resultants may be obtained directly from simple linear static calculations. I.e., it is only the fluctuating part of the response quantities that requires treatment as stochastic processes. It may be shown (see chapter 2.4) that if a zero mean stochastic process is stationary and Gaussian, then its extreme value is proportional to its standard deviation σ r , i.e. k

rk ( x , t )  = k p ⋅ σ rk max

(1.10)

where k p is a time invariant peak factor between about 1.5 and 4.5. Similarly M km a x = M k + k p ⋅ σ M k

where

k = x , y, z

(1.11)

Simple linear static calculations are considered trivial, and thus, the main focus is in the following on the calculation of the standard deviation to fluctuating components, σ rk and σ M k , whether they contain dynamic amplification or not. However, some mention of the calculation of time invariant mean values has been included for the sake of completeness.

Chapter 2 SOME BASIC STATISTICAL CONCEPTS IN WIND ENGINEERING

2.1

Parent probability distributions, mean value and variance

For a continuous random variable X, its probability density function p ( x ) is defined by

P r [x ≤ X ≤ x + d x ] = P ( x + d x ) − P ( x ) =

d P (x ) dx

dx = p (x )d x

(2.1)

where P ( x ) is the cumulative probability function, from which it follows that P r [X ≤ x ] = P (x ) =

x

∫ p (x )dx

(2.2)

−∞

and that lim P ( x ) = 1 . Similarly, for two random variables X and Y the joint probability x →∞

density function is defined by p (x , y ) =

d 2P (x , y )

(2.3)

dx dy

where P ( x , y ) = P r [ X ≤ x ,Y ≤ y ] . The mean value and variance of X are given by x = E [X ] =

+∞

∫

x ⋅ p (x )dx

−∞

Var ( X ) =

σ x2

2 = E ( X − x )  =  

+∞

∫ (x − x )

−∞

2

⋅ p (x )dx

      

(2.4)

14

2 SOME BASIC STATISTICAL CONCEPTS IN WIND ENGINEERING

Equivalent definitions apply to a discrete random variable X . It is in the following assumed that each realisation X k of X has the same probability of occurrence, and thus, the mean value and variance of X may be estimated from a large data set of N individual realisations:

1 N →∞ N

x = lim

Var ( X ) =

      

N

∑Xk

k =1

σ x2

1 = lim N →∞ N

N

∑(X k − x )

2

k =1

(2.5)

The square root of the variance, σ x , is called the standard deviation. Recalling that

E [ X ] = x , the expression for the variance may be further developed into 2 σ x2 = E ( X − x )  = E  X 2 − 2 xX + x 2  = E  X 2  − x 2  

(2.6)

There are three probability density distributions that are of primary importance in wind engineering. These are the Gaussian (normal), Weibull and Rayleigh distributions, each defined by the following expressions:  1  x − x 2  exp  −     2  σx   2πσ x   β β −1  x  x p ( x ) = β β exp  −    γ   γ    1  x 2  x p ( x ) = 2 exp  −    γ  2  γ   p (x ) =

1

           

(2.7)

They are graphically illustrated in Fig. 2.1. It is seen that a Rayleigh distribution is the Weibull distribution with β=2.

2.2 TIME DOMAIN AND ENSEMBLE STATISTICS

Fig. 2.1

2.2

15

Gauss (with x = 0 ) and Weibull distributions

Time domain and ensemble statistics

As mentioned in Chapter 1 there are two types of statistics dealt with in wind engineering: time domain statistics and ensemble statistics. Illustrating time domain statistics, a typical realisation of the outcome of a stochastic process over a period T is illustrated in Fig. 2.2. This may for instance represent a short term recording of the wind velocity at some point in space, or it may equally well represent the displacement response somewhere along the span of the structure. Considering consecutive and for practical purposes equidistant points along the time series as individual random observations of the process, then time domain statistics may be performed on this realisation. It will in the following be assumed that any time domain statistics are based on a continuous or discrete time variable X , which theoretically may attain values between −∞ and +∞ and are applicable over a limited time range between 0 and T, within which the process is stationary and homogeneous (i.e. have constant statistical properties) such that

X = x + x (t )

(2.8)


17

2  T0    2π   1 a2 ⋅ n ⋅ ∫ a sin  t  d t  = n →∞ n ⋅ T  0   2 0  T 0     Given a second zero mean variable comprising two cosine functions with different amplitudes and frequencies, i.e.: x (t ) = a1 ⋅ cos (ω1t ) + a 2 ⋅ cos (ω2t )

σ x2 = lim

where ω1 = 2π T1 and ω2 = 2π T 2 . It is easily seen that if T1 T 2 = 1 then x (t ) = ( a1 + a 2 ) ⋅ cos (ω1t )

and thus, the calculation of its variance is identical to the solution given above, i.e.:

σ x2 =

( a1 + a 2 )2 2

If T1 T 2 ≠ 1 then the variance of x (t ) is given by 1 T →∞ T

σ x2 = lim

T

2 2 2 2 ∫ a1 cos (ω1t ) + 2 a1a 2 cos (ω1t ) cos (ω2t ) + a 2 cos (ω2t ) d t 0

Substituting T = n1 ⋅ T1 into the integration of the first two terms and T = n 2 ⋅ T 2 into the third, where n1 and n 2 are integers, then 1 ⋅ n1 n1 →∞ n T 1 1

σ x2 = lim

+ lim

n 2 →∞

T1

∫

1 ⋅ n1 n1 →∞ n T 1 1

a12 cos 2 (ω1t ) d t + lim

0

1 ⋅ n2 n 2T 2

T1

∫ 2 a1a 2 cos (ω1t ) cos (ω2t ) d t 0

T2

2 2 ∫ a 2 cos (ω2t ) d t 0

It is seen that the first and the third integrals are identical to the integral of a single cosine squared shown above, and thus, they are equal to a12 2 and a 22 2 , respectively. The second integral, containing the product of two cosine functions, may most effectively be solved by the substitution tˆ = ω1t = ( 2π T1 ) t , in which case it is given by 2 a1 a 2  T1 ⋅ T1  2π

2π

∫ 0

T   a a cos tˆ ⋅ cos  1 tˆ d tˆ  = 1 2 π  T 2  

()

  T1  sin 2π 1 − T 2     T1   2 1 −   T2  

  T1    sin 2π 1 +   T 2  +   T1   2 1 +   T 2   

 T  sin  2π 1  aa  T 2  wh ich is = 0 if T1 / T 2 is a n in t eger u n equ a l t o 1 = 1 2  T1 T 2 π ≠ 0 if T1 / T 2 is n ot a n in t eger − T 2 T1

Thus, the variance of x (t ) = a1 ⋅ cos (ω1t ) + a 2 ⋅ cos (ω2t ) is then given by

18


 ( a1 + a 2 )2 2 if ω2 ω1 = 1   a 2 a 2 2 σ x =  1 + 2 if ω2 ω1 is a n in t eger ≠ 1 2 2  a 2 a a sin ( 2π ω2 ω1 ) a 2 + 2 if ω2 ω1 is n ot a n in t eger  1 + 1 2⋅ π ω2 / ω1 − ω1 / ω2 2  2 Similar results would have been obtained if the cosines had been replaced by sinus functions. Thus, if for instance x (t ) = a1 ⋅ cos (ω1t ) + a 2 ⋅ cos (ω2t ) and ω2 ω1 is an integer ≠ 1 , then the variance of x (t ) =

dx = −a1ω1 ⋅ sin (ω1t ) − a 2ω2 ⋅ sin (ω2t ) dt

is given by

σ x2 =

( −a1ω1 )2 + ( −a 2ω2 )2 2

2

= ω12 ⋅

a12 a2 + ω22 ⋅ 2 2 2

Likewise, the variance of the nth derivative of x (t ) , fn (t ) =

σ f2 = ω12 n ⋅

dnx dtn

, is given by

a12 a2 + ω22 n ⋅ 2 2 2

Illustrating ensemble statistics, a situation where N different recordings of a stochastic process within a time window between 0 and T are shown in Fig. 2.3. These may for instance represent N simultaneous realisations of the along wind velocity in space, i.e. they represent the wind velocity variation taken simultaneously and at a certain distance (horizontal or vertical) between each of them. Extracting the recorded values at a given time from each of these realisations will render a set of data X k (t ) , k = 1,...., N . On

this data set ensemble statistics may be performed. This is the type of statistics that provides a stochastic description of the wind field distribution in space. Another example of ensemble statistics is illustrated in Fig. 2.4.a, where the situation is illustrated that N different observations of a stochastic process have been recorded, each taken within a certain time window but in this case not necessarily at the same time. Each of these time series is assumed to be stationary and Gaussian within the short term period that has been considered. In wind engineering this may be an illustration of the situation when a number of time series have been recorded of the wind velocity at a certain point in space, each taken during different weather conditions. In that case one may only be interested in performing statistics on the mean values and discard the rest of the recordings.

20


T

1 Cov x1 x 2 = E  x1 (t ) ⋅ x 2 (t )  = lim T →∞ T

∫ x1 (t ) ⋅ x 2 ( t ) d t

(2.11)

0

Similarly, given two data sets of N individual and equally probable realisations that have been extracted from two random variables, X 1 and X 2 , then the ensemble correlation and covariance are defined by: 1 N →∞ N

R x1 x 2 (τ ) = E [ X 1 ⋅ X 2 ] = lim

N

∑ X 1k ⋅ X 2 k

(2.12)

k =1

Cov x1 x 2 (τ ) = E ( X 1 − x1 ) ⋅ ( X 2 − x 2 ) 

1 N →∞ N

= lim

N

∑ ( X 1k

k =1

)(

− x1 ⋅ X 2 k − x 2

)

(2.13)

However, correlation and covariance estimates may also be taken on the process variable itself. Thus, defining an arbitrary time lag τ , the time domain auto correlation and auto covariance functions are defined by 1 R x (τ ) = E  X (t ) ⋅ X (t + τ )  = lim T →∞ T 1 Cov x (τ ) = E  x (t ) ⋅ x (t + τ )  = lim T →∞ T

T

∫ X (t ) ⋅ X (t + τ ) d t

(2.14)

0

T

∫ x (t ) ⋅ x (t + τ ) d t

(2.15)

0

These are defined as functions because τ is perceived as a continuous variable. As long as τ is considerably smaller than T E  X (t )  = E  X (t + τ )  = x

(2.16)

and thus, the relationship between R x and Cov x is the following Cov x (τ ) = E { X (t ) − x } ⋅ { X (t + τ ) − x } = R x (τ ) − x 2

(2.17)


a)

21

Independent short term realisations

b) The probability of mean values

Fig. 2.4

Ensemble statistics of mean value recordings

There is no reason why τ may not attain negative as well as positive values, and since

E  x (t ) ⋅ x (t − τ )  = E  x (t − τ ) ⋅ x (t )  = E  x (t − τ ) ⋅ x (t − τ + τ )  then

Cov x (τ ) = Cov x ( −τ )

Thus, Cov x is symmetric with respect to its variation with τ .

(2.18)

(2.19)


1 N −j Cov x (τ = j ⋅ ∆t ) = E  x (t ) ⋅ x (t + τ )  = ∑ xk + j ⋅ xk N − j k =1

23

(2.20)

from which it is seen that j must be considerably smaller than N for a meaningful outcome of the auto covariance estimate. The same is true for the auto correlation function in Eq. 2.14.

Example 2.2:

Given a variable: x (t ) = a1 ⋅ sin (ω1t ) , ω1 = 2π T1 . Using the substitutions T = n T1 (where n is an integer) and tˆ = ( 2π T1 ) t , then the auto covariance of x is given by

1 T →∞ T

Cov x (τ ) = lim =

T

1

0

2π a12

2π a12

0

0

2π



T1





0



⋅ n ∫ a1 sin (ω1t ) ⋅ a1 sin (ω1t + ω1τ ) d t  ∫ x (t ) ⋅ x (t + τ ) d t = nlim →∞ n T  1 

∫ sin tˆ ⋅ sin (tˆ + ω1τ ) d tˆ = 2π ∫ sin

2

tˆ ⋅ cos (ω1τ ) + sin tˆ ⋅ cos tˆ ⋅ sin (ω1τ )  d tˆ

2π 2π   2 ˆ ˆ sin (ω1τ ) ˆ tˆ  sin 2td t+ cos (ω1τ ) ∫ sin td ∫ 2   0 0 The first of these integrals is equal to π , while the second is zero, and thus:

=

a12 2π

Cov x (τ ) =

a12 cos (ω1τ ) 2

Since the variance of x (t ) is σ x2 = a12 2 (see example 2.1), then the auto covariance coefficient is given by:

ρ x (τ ) =

Cov x (τ )

σ x2

= cos (ω1τ )

As can be seen: ρ x (τ = 0 ) = 1 .

Similar to the definitions above, cross correlation and cross covariance functions may be defined between observations that have been obtained from two short term realisations X 1 (t ) = x1 + x1 (t ) and X 2 (t ) = x 2 + x 2 (t ) of the same process or alternatively from realisations of two different processes:

1 R X 1 X 2 (τ ) = E  X 1 (t ) ⋅ X 2 (t + τ )  = lim T →∞ T

T

∫ X 1 (t ) ⋅ X 2 (t + τ ) d t 0

(2.21)


26

N  Var  ∑ x i  = E ( x1 + x 2 + ... + x i + ... + x N ) ⋅ x1 + x 2 + ... + x j + ... + x N   i =1 

(

)

N N N  N N ⇒ Var  ∑ x i  = ∑∑ Cov x i ⋅ x j = ∑∑ ρ x i ⋅ x j ⋅ σ i σ j i =1 j =1  i =1  i =1 j =1

(

)

(

)

(2.26)

If x i (t ) are independent (i.e. uncorrelated) then the variance of the sum of the processes is the sum of the variances of the individual processes, i.e. σ 2 wh en i = j if Cov x i ⋅ x j =  x i  0 wh en i ≠ j

(

)

then

N  N Var  ∑ x i  = ∑ σ x2i  i =1  i =1

Example 2.3:

(2.27)

Given an ensemble variable: x = a ⋅ sin (ωt + θ ) , where the probability density distribution of θ 1 for 0 ≤ θ ≤ 2π  is: p (θ ) =  2π 0 elsewh er e 

The ensemble covariance of x k at a time lag τ is then given by

Cov x (τ ) = E  x (t , θ ) ⋅ x (t + τ , θ )  = 2π

=

2π

∫ p (θ ) ⋅ x (t ,θ ) ⋅ x (t + τ ,θ ) dθ 0

1

∫ 2π ⋅ a sin (ωt + θ ) ⋅ a sin (ωt + ωτ + θ ) dθ 0

a2 = 2π

2π

∫ sin (ωt + θ ) ⋅ sin (ωt + θ ) ⋅ cos (ωτ ) + cos (ωt + θ ) ⋅ sin (ωτ ) dθ 0

2π  sin (ωτ ) 2π a2  2 = sin 2 (ωt + θ ) d θ  cos (ωτ ) ∫ sin (ωt + θ ) d θ + ∫ 2π  2 0 0 

which, after the substitution θˆ = ωt + θ , renders

2.3 THRESHOLD CROSSING AND PEAKS

Cov x (τ ) =

a2 cos (ωτ ) 2

ωt + 2 π

∫

sin 2 θˆd θˆ +

sin (ωτ ) ωt + 2π

ωt

2

∫

27

sin 2θˆd θˆ

ωt

As shown in example 2.2, the first of these integrals is equal to π , while the second is zero, and thus: Cov x (τ ) =

a2 cos (ωτ ) 2

There are still other types of time domain and ensemble statistics that are of great importance in wind engineering and that have not yet been mentioned. These comprise the properties of threshold crossing, the distributions of peaks and extreme values, and finally, the auto and cross spectral densities, which are frequency domain properties of the process, i.e. they are frequency domain counterparts to the concepts of variance and covariance. These are dealt with below.

2.3

Threshold crossing and peaks

In Fig. 2.8 is illustrated a time series realisation x (t ) of a Gaussian stationary and homogeneous process (for simplicity with zero mean value), taken over a period T. First we seek to develop an estimate of the average frequency fx ( a ) between the events that x (t ) is crossing the threshold a in its upward direction. Let a single upward crossing take place in a time interval ∆t that is small enough to justify the approximation

x (t + ∆t ) ≅ x (t ) + x (t ) ⋅ ∆t

(2.28)

The probability of an up crossing event during ∆t is then given by

P  x (t ) ≤ a a n d x (t ) + x (t ) ⋅ ∆t > a  = fx ( a ) ⋅ ∆t

(2.29)

from which it follows that ∞ a  1   ∫ p xx ( x , x ) d x  d x ∫ ∆t →0 ∆t  0  a − x ⋅∆t 

fx ( a ) = lim

(2.30)

29

2.3 THRESHOLD CROSSING AND PEAKS

fx ( a ) =

 1  a 2  ∞  1  x 2  1  −  exp  −  x ⋅ exp    d x  2  σ x   ∫0  2  σ x   2πσ x 2πσ x    

⇒ fx ( a ) =

1

 1  a 2   1  a 2  1 σ x  −  f 0 exp ⋅ ⋅ exp  −  = ⋅    x ( ) 2π σ x  2 σx    2 σx       fx ( 0 ) =

where:

1 σ x ⋅ 2π σ x

(2.33)

(2.34)

is the average zero up–crossing frequency of the process (see Eq. 2.95). If x (t ) is also narrow banded, such that a zero up crossing and a peak x p (larger than zero) are simultaneous events (as shown for the process in Fig. 2.8), then the expected number of

( )

peaks x p > a p is fx a p ⋅ T , while the total number of peaks is fx ( 0 ) ⋅ T . Thus

( )

P r  x p ≤ a p  = P a p = 1 −

( )

fx a p

(2.35)

fx ( 0 )

from which it follows that the probability density distribution to a p is given by

( )

p ap =

( )  = −

fx a p d d  1 − P ap = dap dap  fx ( 0 ) 

( )

( )

⇒ p ap =

 1  a 2  −  p   exp 2  2 σx   σx   ap

( )

Thus, the probability density p a p



1

fx ( 0 )

⋅

( )

d fx a p dap

(2.36)

of peaks to a narrow banded Gaussian process is a

Rayleigh distribution (see Eq. 2.7). The distribution is illustrated on the right hand side of Fig. 2.8 (see also Fig. 2.1).

2.4 EXTREME VALUES

31

values, as illustrated in Fig. 2.9.b. Referring to Eq. 2.33 and Fig. 2.8, an extreme peak value a p = x m a x within each short term realisation occur when

( )

 fx a p    Let therefore

−1

→T

(2.37)

κ = fx ( x m a x ) ⋅ T

(2.38)

be an ensemble variable signifying the event that x ( 0 ≤ t ≤ T

)

exceeds a given value

x m a x . The probability that κ occurs only once within each realisation is an event that

coincides with the occurrence of x m a x , i.e. they are simultaneous events. They are rare events at the tail of the peak distribution given in Eq. 2.36, and for the statistics of such events it is a reasonable assumption that they will also comply to an exponential distribution, i.e. that Pκ (1,T ) = Px m a x ( x m a x T ) = exp ( −κ )

(2.39)

Introducing Eqs. 2.33 (with a = x m a x ) into 2.38 and solving for x m a x , then the following is obtained

{

x m a x = σ x ⋅ 2 ⋅ ln  fx ( 0 ) T  − 2 ⋅ ln κ

1

}2

 ln κ ≈ σ x ⋅ 2 ⋅ ln  fx ( 0 ) ⋅ T  1 −  ⋅ 2 ln  fx ( 0 ) ⋅ T  where the approximation

(1 − x )n

≈1 −n ⋅x

   

(2.40)

has been applied, assuming that

ln  fx ( 0 ) ⋅ T  is large as compared to ln κ . Thus, observing that x m a x = 0 corresponds

to κ = ∞ , while x m a x = ∞ corresponds to κ = 0 , the mean value of x m a x may be estimated from ∞  d Px m a x xmax = ∫ xmax ⋅   dx max  0

∞  d x m a x = ∫ x m a x  0

0

∞

∞

0

 d Px m a x ⋅  dκ 

  dκ   ⋅  d x m a x   d xmax 

= ∫ x m a x ⋅  − exp ( −κ )  d κ = ∫ x m a x ⋅ exp ( −κ ) d κ

2.5 AUTO SPECTRAL DENSITY

33

Given a stochastic variable X (t ) = x + x (t ) , the expected value of its largest peak during a realisation with length T may then be estimated from X max = x + k p ⋅σ x

(2.44)

where the peak factor k p is given by

k p = 2 ⋅ ln  fx ( 0 ) ⋅ T  +

γ 2 ⋅ ln  fx ( 0 ) ⋅ T 

(2.45)

For fairly broad banded processes this peak factor will render values between 2 and 5. Plots of k p and σ x m a x σ x are shown in Fig. 2.10. It should be acknowledged that when x (t ) becomes ultra narrow banded then k p → 2 , because for a single harmonic component x (t ) = cx ⋅ cos (ωx t ) , 0 < t < T

(2.46)

the variance

1 T →∞ T

σ x2 = lim

T

∫ cx ⋅ cos (ωx t )

2

dt

0

1 = lim ⋅n ⋅ n →∞ n ⋅ T x

Tx

∫ 0

2

  2π   c2 t  d t = x cx ⋅ cos  2   T x  

(2.47)

and thus, for such a process x m a x = cx = σ x ⋅ 2 . Therefore, Eq. 2.45 is only applicable for fairly broad banded processes.

2.5

Auto spectral density

The auto spectral density contains the frequency domain properties of the process, i.e. it is the frequency domain counterpart to the concept of variance. The various steps in the development of an auto spectral density function are illustrated in Fig. 2.11. Given a zero mean time variable x (t ) with length T and performing a Fourier transformation of x (t ) implies that it may be approximated by a sum of harmonic components X k (ωk , t ) , i.e.

34


N

∑ X k (ωk , t ) N →∞

x (t ) = lim

where

k =1

 ωk = k ⋅ ∆ω   ∆ω = 2π / T

(2.48)

The harmonic components in Eq. 2.48 are given by

X k (ωk , t ) = ck ⋅ cos (ωk t + ϕk )

(2.49)

where the amplitudes ck = a k2 + bk2 and phase angles ϕk = a r c t a n (bk / a k ) , and where the constants a k and bk are given by a k  2 b  =  k T

 cos ωk t 

T

∫ x (t ) sin ωk t  d t

(2.50)

0

As shown in Fig. 2.11 the auto-spectral density of x (t ) is intended to represent its variance density distribution in the frequency domain. Hence, the definition of the single-sided auto-spectral density S x associated with the frequency ωk is E  X k2  σ X2 k = S x (ωk ) = ∆ω ∆ω

(2.51)

which, when T becomes large, is given by 1 1 ⋅ T →∞ ∆ω T

S x (ωk ) = lim

T

∫ ck cos (ωk t + ϕk )

2

dt

(2.52)

0

Introducing the period of the harmonic component, T k = 2π / ωk , and replacing T with n ⋅ T k , n → ∞ , then the following is obtained

1 1 ⋅ ⋅n ⋅ n →∞ ∆ω n ⋅ T k

S x (ωk ) = lim

Tk

∫ 0

2

  2π  ck2 ⋅ + = c t d t cos ϕ  k  k  2 ∆ω  Tk   

(2.53)

36


In Fig. 2.11, the arrival at S x (ωk ) is shown via the amplitude spectrum (or the Fourier amplitude diagram) to ease the understanding of the concept of spectral density representations. It is seen from this illustration that it is not possible to retrieve the parent time domain variable from the spectral density function alone, because it does not contain the necessary phase information (unless a corresponding phase spectrum is also established). From its very definition the spectrum contains information about the variance distribution in frequency domain, and from Eqs. 2.51 and 2.53 it is seen that N

∑ σ X2 k N →∞

σ x2 = lim

k =1

c2

N

∑ 2k N →∞

= lim

N

∑ S x (ωk ) ⋅ ∆ω N →∞

= lim

k =1

(2.54)

k =1

In a continuous format, i.e. in the limit of both N and T approaching infinity, the singlesided auto-spectral density is defined by

E  X 2 (ω , t )  S x (ω ) = lim lim T →∞ N →∞ ∆ω

(2.55)

where X (ω , t ) is an arbitrary Fourier component of x (t ) . In the limit ∆ω → d ω , and thus, the variance of the process may be calculated from ∞

σ x2 = ∫ S x (ω ) d ω

(2.56)

0

The development above may more conveniently be expressed in a complex format. Adopting a frequency axis spanning the entire range of both positive and (imaginary) negative values, introducing the Euler formulae

 eiωt  1 i  cos ωt   =   e −iωt  1 −i  sin ωt 

(2.57)

(where i = −1 ) and defining the complex Fourier amplitude

dk = then:

1 ( a k − i ⋅ bk ) 2

∞

∞

−∞

−∞

x (t ) = ∑ X k (ωk , t ) = ∑ d k (ωk ) ⋅ ei ⋅ωk t

(2.58) (2.59)

2.5 AUTO SPECTRAL DENSITY

37

Taking the variance of the complex Fourier components in Eq. 2.59 and dividing by ∆ω ,

E  X k* ⋅ X k  1  = T ∆ω

T

∫

(d e

* −iωk t k

)(d e ω ) d t = d d k

i kt

∆ω

0

* k k

∆ω

(2.60)

which may be further developed into

E  X k* ⋅ X k  1 ( a k + i ⋅ bk ) ⋅ ( a k − i ⋅ bk ) c2 ⇒ = = k ∆ω ∆ω 4 4 ∆ω

(2.61)

It is seen (see Eq. 2.53) that this is half the auto spectral value associated with ωk . Thus, a symmetric double-sided auto spectrum associated with −ωk as well as +ωk may be defined with a value that is half the corresponding value of the single sided autospectrum. Extending the frequency axis from minus infinity to plus infinity and using the complex Fourier components X k given in Eq. 2.59 above, this double sided auto spectrum is then defined by S x ( ±ωk ) =

E  X k* ⋅ X k  d * d c2 = k k = k ∆ω ∆ω 4 ∆ω

(2.62)

which, in the limit of T a n d N → ∞ , becomes the continuous function S x ( ±ω ) , and from which the variance of the process may be obtained by integration over the entire positive as well as negative (imaginary) frequency range

σ x2 =

+∞

∫ S x ( ±ω ) d ω

(2.63)

−∞

Thus, the connection between double- and single-sided spectra is simply that S x (ω ) = 2 ⋅ S x ( ±ω ) . Assuming that the process is stationary and of infinite length, such that the position of the time axis for integration purposes is arbitrary, then it is in the literature of mathematics usually considered convenient to introduce a non-normalized amplitude (which may be encountered in connection with the theory of generalised Fourier series and identified as a Fourier constant)

a k (ωk ) =

T

∫ x (t ) ⋅ e

0

−i ⋅ωk t

d t = T ⋅ dk

(2.64)

38


in which case the double-sided auto-spectral density associated with ±ωk is defined by

S x ( ±ωk ) =

(

)

a k* / T ⋅ ( a k / T d k* ⋅ d k = 2π / T ∆ω

)

=

1 ⋅ a k* a k 2π T

(2.65)

In the limit of T a n d N → ∞ this may be written on the following continuous form

1 ⋅ a * (ω ) ⋅ a (ω ) T →∞ N →∞ 2π T

S x ( ±ω ) = lim lim

(2.66)

and accordingly, the single sided version is given by S x (ω ) = lim

1

T →∞ π T

⋅ a * (ω ) ⋅ a (ω )

(2.67)

where it is taken for granted that N is sufficiently large. The auto-spectral density S x (ω ) is defined by use of circular frequency ω as shown above. It may be replaced by a corresponding definition S x ( f ) using frequency f (with unit H z = sek −1 ). Since

S x (ω ) ⋅ ∆ω and S x ( f ) ⋅ ∆f both represent the variance of the process at ω and f , they must give the same contribution to the total variance of the process, and thus

S x ( f ) ⋅ ∆f = S x (ω ) ⋅ ∆ω = S x (ω ) ⋅ ( 2π ⋅ ∆f ) 2 * ⋅a (f )⋅a (f ) T →∞ N →∞ T

⇒ S x ( f ) = 2π ⋅ S x (ω ) = lim lim

2.6

(2.68)

Cross-spectral density

The cross spectral density contains the frequency domain and coherence properties between processes, i.e. it is the frequency domain counterpart to the concept of covariance. Given two stationary time variable functions x (t ) and y (t ) , both with length T and zero mean value (i.e. E  x (t )  = E  y (t )  = 0 ), and performing a Fourier

transformation (adopting a double-sided complex format) implies that x (t ) and y (t ) may be represented by sums of harmonic components X k ( ωk , t) and Y k (ωk , t ) , i.e.

2.6 CROSS SPECTRAL DENSITY

39

N

 x (t )   X k (ωk , t )    = lim ∑    y (t )  N →∞ − N  Y k (ωk , t ) 

(2.69)

where:  X k (ωk , t )  1 a X k (ωk )  i ⋅ω t  ⋅ e k and  =   Y k (ωk , t )  T  aY k (ωk ) 

T /2 a X (ωk )   x (t )  −i ⋅ω t  k  = lim ∫   ⋅ e k dt  aY k (ωk )  T →∞ −T / 2  y (t ) 

and where ωk = k ⋅ ∆ω and ∆ω = 2π /T . The definition of the double-sided crossspectral density S xy associated with the frequency ωk is then

S xy ( ±ωk ) =

E  X k* ⋅ Y k  1 = a X* k ⋅ aY k ∆ω 2π T

(2.70)

Since the Fourier components are orthogonal

S xy (ωk , t ) ⋅ ∆ω wh en i = j = k E  X i (ωi , t ) ⋅ Y j ω j , t  =  0 wh en i ≠ j

(

)

(2.71)

it follows from Eqs. 2.69 and 2.70 that an estimate of the covariance between x (t ) and y (t ) are given by N  N  N  Cov xy = E  x (t ) ⋅ y (t )  = lim E  ∑ X i  ⋅  ∑ Y j   = lim ∑ ( E [ X k ⋅ Y k ]) N →∞   − N   N →∞ − N  − N N

∑ S xy ( ±ωk ) ⋅ ∆ω N →∞

⇒ Cov xy = lim

(2.72)

−N

In a continuous format, i.e. in the limit of both N and T approaching infinity, the doublesided cross-spectral density is defined by

40


E  X * (ω , t ) ⋅ Y (ω , t )  S xy ( ±ω ) = lim lim T →∞ N →∞ ∆ω 1 * a X ( ω ) ⋅ aY ( ω ) = lim lim T →∞ N →∞ 2π T

(2.73)

The single sided version is then simply

1

S xy (ω ) = 2 ⋅ S xy ( ±ω ) = lim lim

T →∞ N →∞ π T

a X* (ω ) ⋅ aY (ω )

(2.74)

while the corresponding single-sided version using frequency f (Hz), is defined by

S xy ( f ) = 2π ⋅ S xy (ω ) = lim lim

2

T →∞ N →∞ T

⋅ a x* ( f ) ⋅ a y ( f )

(2.75)

Thus, the covariance between the two processes may be calculated from +∞

Cov xy =

∫

−∞

∞

∞

0

0

S xy ( ±ω ) d ω = ∫ S xy (ω ) d ω = ∫ S xy ( f ) d f

(2.76)

The cross-spectrum will in general be a complex quantity. With respect to the frequency argument, its real part is an even function labelled the co–spectral density Coxy (ω ) , while its imaginary part is an odd function labelled the quad–spectrum Qu xy (ω ) , i.e.

S xy (ω ) = Coxy (ω ) − i ⋅ Qu xy (ω )

(2.78)

as illustrated in Fig. 2.12. Alternatively, S xy (ω ) may be expressed by its modulus and phase, i.e.

S xy (ω ) = S xy (ω ) ⋅ e

i ⋅ϕxy (ω )

where the phase spectrum ϕ xy (ω ) = a r c t a n Qu xy (ω ) Coxy (ω )  .

(2.79)

2.8 COHERENCE FUNCTION AND NORMALIZED CO-SPECTRUM

43

This shows that the auto spectral density is the Fourier transform of the auto covariance function. Vice versa, it follows that the auto covariance function, which is the Fourier constant to the spectral density, is given by

Cov x (τ ) =

+∞

∫ S x (ω ) ⋅ e

iωτ

dω

(2.84)

−∞

Similarly, the cross covariance function together with the cross spectral density will also constitute a pair of Fourier transforms:

S xy (ω ) =

2.8

1 2π

+∞

∫

Cov xy (τ ) ⋅ e −iωτ dτ and

Cov xy (τ ) =

−∞

+∞

∫ S xy (ω ) ⋅ e

iωτ

d ω (2.85)

−∞

Coherence function and normalized co-spectrum

The coherence function is defined by

Coh xy (ω ) =

S xy (ω )

2

(2.86)

S x (ω ) ⋅ S y (ω )

If x (t ) and y (t ) are realisations of the same process, then S x (ω ) = S y (ω ) and the cross-spectrum S xy (ω ) = S xx (ω ) is given by

S xx (ω ) = S x (ω ) ⋅ Coh xx (ω ) ⋅ e

iϕxx (ω )

(2.87)

Coh xx (ω ) is called the root–coherence function and ϕxx is the phase spectrum (see Eq. 2.79) . In the practical use of cross-spectra all imaginary parts will cancel out, and thus it is only the co-spectrum that is of interest. Therefore, a normalised co-spectrum is defined

ˆ Co xx (ω ) =

Re S xy (ω ) 

S x (ω ) ⋅ S y (ω )

(2.88)

44


Again, if x (t ) and y (t ) are realisations of the same stationary and ergodic process, then S x (ω ) = S y (ω ) and the real part of the cross-spectrum is given by

ˆ Re S xy (ω )  = S x (ω ) ⋅ Co xx ( ω )

2.9

(2.89)

The spectral density of derivatives of processes

It may in some cases be of interest to calculate the spectral density of the time derivatives [e.g. x (t ) and x (t ) ] of processes. In structural engineering this is particularly relevant if x (t ) is a response displacement of such a character that it is necessary to evaluate as to whether or not it is acceptable with respect to human perception, in which case the design criteria most often will contain acceleration requirements. Since (see Eq. 2.59) ∞

∞

−∞

−∞

∞

∞

−∞

−∞

x (t ) = ∑ X k (ωk , t ) = ∑ iωk ⋅ d k (ωk ) ⋅ ei ⋅ωk t x (t ) = ∑ Xk (ωk , t ) = ∑ ( iωk ) ⋅ d k (ωk ) ⋅ ei ⋅ωk t 2

(2.90) (2.91)

and the double sided spectral density in general is given by the complex Fourier amplitude multiplied by its conjugated counterpart (see Eq. 2.62), then *

iωk d k (ωk )  ⋅ iωk d k (ωk )  d *d = ωk2 k k = ωk2 S x ( ±ωk ) S x ( ±ωk ) =  ∆ω ∆ω

(2.92)

*

( iω )2 d (ω )  ⋅ ( iω )2 d (ω )  * k k  k k   k   k  = ω 4 d k d k = ω 4 S ±ω S x ( ±ωk ) =  k k x ( k ) (2.93) ∆ω ∆ω In the limit of T a n d N → ∞ this may be written on the following continuous form

S x ( ±ω )  ω 2    =  4  ⋅ S x ( ±ω ) S x ( ±ω )  ω 

(2.94)

2.10 SPATIAL AVERAGING IN STRUCTURAL RESPONSE CALCULATIONS

45

Thus, the spectral density of time derivatives of processes may be obtained directly from the spectral density of the process itself. Since the single sided spectrum is simply twice the double sided, Eq. 2.94 will also hold if S x ( ±ω ) , S x ( ±ω ) and S x ( ±ω ) are replaced by S x (ω ) , S x (ω ) and S x (ω ) . From S x (ω ) and S x (ω ) the average zero crossing frequency fx ( 0 ) of the process x (t ) may be found. Referring to Eq. 2.34, 2.56 and introducing S x (ω ) = ω 2 S x (ω ) , the

following applies: 1/2

fx ( 0 ) =

1 σ x ⋅ 2π σ x

∞ 2   ∫ ω S x (ω ) d ω  1 0  = ⋅ ∞  2π  S (ω ) d ω   ∫ x   0 

=

µ2 1 ⋅ 2π µ0

(2.95)

where for convenience the so–called n th spectral moment ∞

µn = ∫ ω n ⋅ S x ( ω ) d ω

(2.96)

0

has been introduced.

2.10

Spatial averaging in structural response calculations

A typical situation in structural engineering is illustrated in Fig. 2.14. A cantilevered tower–like beam is subject to a fluctuating short term (stationary) and distributed wind load. The problem at hand is to predict a load effect, e.g. the bending moment at the base. It is for simplicity assumed that the beam is so stiff that it is not necessary to include any dynamic amplification. It is taken for granted that the wind load may be split into a mean and a fluctuating part, i.e.

q ytot = q y ( x ) + q y ( x , t )

(2.97)

where q y ( x ) is a deterministic quantity and q y ( x , t ) is a stochastic variable. Correspondingly, the load effect is split into a mean and a fluctuating part M tot = M + M (t )

(2.98)


1 T →∞ T

2 σM = lim LL

=

∫ ∫ GM 0 0

47

L  L  , G x q x t d x ⋅ ⋅  ( ) ( ) 1  ⋅  ∫ G M ( x 2 ) ⋅ q y ( x 2 ,t ) ⋅ d x 2  d t ∫ ∫ M 1 y 1   0  0 0 (2.101) T   1 ( x1 ) ⋅ G M ( x 2 ) ⋅ Tlim ∫ q y ( x1 , t ) ⋅ q y ( x 2 , t ) d t  d x1 d x 2  →∞ T 0 

T

Recalling that the cross covariance function of q y ( x , t ) is given by 1 T →∞ T

T

1 = lim T →∞ T

T

Cov q y q y ( ∆x ,τ = 0 ) = lim

∫ q y ( x , t ) ⋅ q y ( x + ∆x , t ) d t 0

(2.102)

∫ q y ( x1 , t ) ⋅ q y ( x 2 , t ) d t = Covq y ( ∆x ) 0

where the separation ∆x = x 2 − x1 , and introducing the covariance coefficient

ρq y ( ∆x ) = Cov q y ( ∆x ) / σ q2y , it is seen that Eq. 2.101 simplifies into LL

2 σM = σ q2y ⋅ ∫ ∫ G M ( x1 ) ⋅ G M ( x 2 ) ⋅ ρq y ( ∆x ) d x1 d x 2

(2.103)

0 0

The square root of the double integral 1/2

JM

L L  =  ∫ ∫ G M ( x1 ) ⋅ G M ( x 2 ) ⋅ ρq y ( ∆x ) d x1 d x 2   0 0 

(2.104)

is in wind engineering often called the joint acceptance function, because it contains the necessary statistical (i.e. variance) averaging in space. Thus,

σ M = σqy ⋅ J M

(2.105)

Similarly, the auto spectral density of M (t ) may be obtained by taking the Fourier transform on either side of Eq. 2.99 L

a M (ω ) = ∫ G M ( x ) ⋅ a q y ( x , ω ) ⋅ d x 0

and applying Eq. 2.67

(2.106)


48

1 * ⋅ aM (ω ) ⋅ a M (ω ) T →∞ π T L  L  1  = lim ⋅  ∫ G M ( x ) ⋅ a q* y ( x , ω ) ⋅ d x  ⋅  ∫ G M ( x ) ⋅ a q y ( x , ω ) ⋅ d x     T →∞ π T  0  0 

S M (ω ) = lim

LL

=



1

∫ ∫ G M ( x1 ) ⋅ G M ( x 2 ) ⋅ Tlim →∞ π T 0 0

⇒ S M (ω ) =

 ⋅ a q* y ( x1 , ω ) ⋅ a q y ( x 2 , ω )  d x1 d x 2 

LL

∫ ∫ G M ( x 1 ) ⋅ G M ( x 2 ) ⋅ S q y ( ∆x , ω ) d x 1 d x 2

(2.107)

0 0

where S q y ( ∆x , ω ) is the cross spectral density of the fluctuating part q y ( x , t ) of the distributed load, and ∆x = x 2 − x1

is spatial separation. Integrating over the entire

frequency domain will then render the variance of M (t ) : ∞

∞ L L



 0 0



2 σM = ∫ S M (ω ) d f = ∫  ∫ ∫ G M ( x1 ) ⋅ G M ( x 2 ) ⋅ S q y ( ∆x , ω ) d x1 d x 2  dω 0

=

σ q2y

0

∞ L L

⋅ ∫  ∫ ∫ G M ( x1 ) ⋅ G M ( x 2 ) ⋅ Sˆ q y 0 0 0

 ( ∆x , ω ) d x1 d x 2  d ω 

(2.108)

where σ q y is defined above and Sˆ q y ( ∆x , ω ) is the normalised (but not non– dimensional) version of S q y ( ∆x , ω ) , i.e. Sˆ q y ( ∆x , ω ) = S q y ( ∆x , ω ) / σ q2y

(2.109)

Introducing xˆ = x / L and correspondingly ∆xˆ = xˆ1 − xˆ 2 , then a normalised frequency

domain version of the joint acceptance function may be defined by 1/2

1 1  Jˆ M (ω ) =  ∫ ∫ G M ( xˆ1 ) ⋅ G M ( xˆ 2 ) ⋅ Sˆ q y ( ∆xˆ , ω ) d xˆ1 d xˆ 2   0 0 

(2.110)

in which case the following is obtained: 1/2

σ M = σqy

∞ 2  ⋅ L ⋅  ∫ Jˆ M (ω ) d ω   0 

(2.111)


50

M (t ) =

N

∑ M k (t )

(2.114)

k =1

Its variance is then given by (see Eq. 2.100)

2 σM

1 = lim T →∞ T

T

1 T →∞ T

T

= lim

2 ⇒ σM =

1 ∫  M (t ) d t = Tlim →∞ T 0 2

2

T

N  ∫ k∑=1 M k (t ) d t 0

∫ [ M 1 + ..... + M k + ..... + M N ] ⋅ [ M 1 + ..... + M k + ..... + M N ] d t 0

N

N

1

T

 M n (t ) ⋅ M m (t )  d t ∑ ∑ Tlim →∞ T ∫ 

n =1 m =1

(2.115)

0

As can be seen, the transition from a single summation to a double summation is necessary to capture all the cross products. Introducing Eqs. 2.112 and 2.113, the following is obtained: 2 = σM

N

N

T

1 

L

G M ( x n ) q y ( x n ,t ) ∑ ∑ Tlim →∞ T ∫  N 

n =1 m =1

0

⋅ G M ( x m ) q y ( x m ,t )

L dt N 

T  N N   L 2 1 =  ∑ ∑ G M ( x n ) ⋅ G M ( x m ) ⋅ lim ∫ q y ( x n , t ) ⋅ q y ( x m , t )  d t  ⋅   T →∞ T 0 n =1 m =1   N 

⇒

2 σM

 σqy ⋅ L =  N 

2

  N N   ⋅  ∑ ∑ G M ( x n ) ⋅ G M ( x m ) ⋅ ρq ( ∆x )  y  n =1 m =1  

(2.116)

where ρq y ( ∆x ) is the covariance coefficient to the distributed load, and where

∆x = x m − x n . The expression in Eq. 2.116 is equivalent to that which was obtained in Eq. 2.103.

Example 2.4: Considering the cantilevered beam shown in Fig. 2.15, then the reduced variance of the base moment fluctuations is given by:


 σ  M  σ q L2  y

2

  = 1  N2 

N

N

∑ ∑ Gˆ M ( xˆn ) ⋅ Gˆ M ( xˆm ) ⋅ ρq y ( ∆xˆ )

n =1 m =1

where Gˆ M ( xˆ ) = G M L = x L = xˆ , ∆x = x m − x n

(

covariance coefficient ρq y ( ∆x ) = exp − ∆x

ρq y ( ∆x ) = exp ( −∆xˆ ) .

51

Choosing

a

z

Lu

)

reduced

and ∆xˆ = ∆x L . Assuming that the

and setting for simplicity integration

increment

corresponding position vector xˆ = [0.1 0.3 0.5 0.7 0.9 ]

T

z

L u = L , then

∆x L = 0.2

then the influence function

multiplications Gˆ M ( xˆ n ) ⋅ Gˆ M ( xˆ m ) are given by Gˆ M ( xˆ n ) ⋅ Gˆ M ( xˆ m ) :

Gˆ M ( xˆ m )

0.1 0.3 0.5 0.7 0.9

Gˆ M ( xˆ n ) 0.1

0.3

0.5

0.7

0.9

0.01 0.03 0.05 0.07 0.09

0.03 0.09 0.15 0.21 0.27

0.05 0.15 0.25 0.35 0.45

0.07 0.21 0.35 0.49 0.63

0.09 0.27 0.45 0.63 0.81

The covariance coefficient ρq y ( ∆xˆ ) is given by:

ρq y ( ∆xˆ ) :

xˆ m

0.1 0.3 0.5 0.7 0.9

xˆ n 0.1

0.3

0.5

0.7

0.9

1 0.82 0.67 0.55 0.45

0.82 1 0.82 0.67 0.55

0.67 0.82 1 0.82 0.67

0.55 0.67 0.82 1 0.82

0.45 0.55 0.67 0.82 1

The inner product Gˆ M ( xˆ n ) ⋅ Gˆ M ( xˆ m ) ⋅ ρq y ( ∆xˆ ) is then:

and


52 n

1

2

m

3

4

5

Gˆ M ( xˆ n ) ⋅ Gˆ M ( xˆm ) ⋅ ρq y ( ∆xˆ )

1 2 3 4 5

0.01 0.025 0.034 0.039 0.041

0.025 0.09 0.123 0.141 0.149

0.034 0.123 0.25 0.287 0.302

0.039 0.141 0.287 0.49 0.517

0.041 0.149 0.302 0.517 0.81

As can be seen, the inner product Gˆ M ( xˆ n ) ⋅ Gˆ M ( xˆm ) ⋅ ρq y ( ∆xˆ ) is symmetric about the diagonal m = n and increasing with increasing distance from the base of the beam. Thus, the reduced variance of the base moment fluctuations is given by:

 σ  M  σ q L2  y

2

  = 1 [0.01 + 0.09 + 0.25 + 0.49 + 0.81 + 2 ⋅ ( 0.025 + 0.034 + 0.039 + 0.041  52  +0.123 + 0.141 + 0.149 + 0.287 + 0.302 + 0.517 )  ≈ 0.2 ⇒ σ M ≈ 0.45 ⋅ σ q y ⋅ L2

Chapter 3 STOCHASTIC DESCRIPTION OF TURBULENT WIND

The description of the wind field given below is only intended to provide the theoretical basis that is necessary for the ensuing calculations of structural response. More comprehensive descriptions have been presented by Simiu & Scanlan [4] and by Dyrbye & Hansen [5], where guidelines with respect to the choice of typical input parameters to the stochastic description of the wind field may be found. Such information has also been given by Solari & Piccardo [6]. The most comprehensive source of wind engineering data is provided by Engineering Science Data Unit [7]. Basic theory of turbulence may be found in many text books, see e.g. Batchelor [8] and Tennekes & Lumley [9]. As shown in Fig. 1.3.a the wind velocity vector at a certain point is described by its components (see Eqs. 1.2 – 1.4) in the Cartesian coordinate system ( x , y , z ) f with x f in the main flow direction and z f in the vertical direction. It is taken for granted that the wind field met by the structure is stationary and homogeneous within the time and space that is considered. A statistical description of the wind field comprises three levels: the long term variation of the mean wind velocity, the short term single point time domain variation of the turbulence components, and finally, the short term spatial distribution of the turbulence components.

3.1

Mean wind velocity

The statistical properties of the mean wind velocity V ( z f ) are required in order to establish a basis for the calculation of structural design load effects during the weather conditions that have been deemed representative for the purpose of obtaining sufficient safety against structural failure. A design check with respect to ultimate structural strength will only require information regarding the wind field properties under a characteristic extreme weather condition, but the properties under several representative weather conditions are required if vortex shedding may occur. If a fatigue design check is relevant even further information is required with respect to the wind climate on the construction site. Thus, mean wind statistics must be based on data covering numerous

54

3 STOCHASTIC DESCRIPTION OF TURBULENT WIND

( )

meteorological observations over several years, as it is the values of V z f

under a

large variation of weather conditions that are of interest (or ideally under any possible weather condition at the site in question). Such statistics are usually performed on the mean wind velocity at z f = 10 m and averaged over a period of T = 10 m in . A

typical instantaneous wind velocity profile in the main flow direction is illustrated on the left hand side of Fig. 3.1, together with the mean velocity and turbulence variation with zf .

Fig. 3.1

The wind velocity and turbulence profiles

A theoretical approach renders a natural logarithmic profile for the height variation of the mean wind velocity (first shown by Millikan [30])

  zf   kT ⋅ ln   wh en z f > z m in V10 z f   z0  = V10 (10 )   z m in  kT ⋅ ln  z  wh en z f ≤ z m in  0  

( )

(3.1)

3.1 MEAN WIND VELOCITY

55

where the index 10 has been added to V, indicating an averaging period of 10 minutes, while kT , z 0 and z m in are parameters characteristic to the terrain in question. The height z m in has been introduced because such a velocity profile has a limited validity close to the ground, where turbulence and directional effects prevail. z 0 is usually called the roughness length. It coincides with the height at which the velocity variation according to Eq. 3.1 is zero. Typical values of kT and z 0 varies from about 0.15 and 0.01 for open sea and countryside without obstacles to about 0.25 and 1.0 for built up urban areas. Corresponding values of z m in varies between 2 and about 15 m. (Other profiles, e.g. the power law profile, may be found in the literature.) Any statistical properties related to the mean wind velocity is in the following

(

)

associated with V10 z ref , where z ref is a chosen reference height. In general, z ref = 10 m as mentioned above, but for a bridge whose main girder is located at a

certain height above the sea or terrain, z ref will often be chosen at this height. To

(

simplify notations V10 z ref

)

is set equal to V r or V a for the remaining part of this

chapter. The indexes r and a indicate whether the relevant statistical calculations have been performed on the parent population or on a reduced population of annual maxima. Data from a large population of parent observations may usually be fitted to a Weibull distribution, i.e. the cumulative and corresponding density distributions are given by γ (ϕ )    V    Pr (V r ≤ V , ϕ ) = 1 − α (ϕ ) ⋅ exp −      β (ϕ )  

     γ (ϕ ) −1 γ (ϕ )   d Pr α (ϕ ) ⋅ γ (ϕ )  V    V   p r (V , ϕ ) = = ⋅ ⋅ exp −     dV β (ϕ )    β (ϕ )   β (ϕ )  

(3.2)

where ϕ is the main flow direction and α (ϕ ) and β (ϕ ) are parameters to be fitted to the relevant data. If the directionality effect is omitted, i.e. for omni-directional wind, the data may usually be fitted to a Rayleigh density distribution  1  V 2  −  ⋅ exp (3.3)    2  Vm   V m2   is the velocity at the apex of the distribution, as illustrated in Fig. 3.2. Thus,

p r (V ) =

where V m

V

the probability µ of exceeding a certain limiting value V s (see Fig. 3.2) is given by

3.1 MEAN WIND VELOCITY

where α = qV − γ ⋅ β and β =

(

57

)

6 / π ⋅ σ q , and where qV is the mean value of the

velocity pressure recordings, σ q is the corresponding standard deviation and γ ≈ 0.5772 is the Euler constant. α and β are parameters that are characteristic to the distribution of the recorded data. If the return period R p is defined as the average number of years between rare qV a events, then a small probability µ of exceeding a certain limiting design value qV d

(

)

(

)

µ = Pa qV a > qVd = 1 − Pa qV a ≤ qV d ≈

1 Rp

(3.6)

and thus, the qV d that corresponds to such a return period is given by

1−

  qV − α   1 = Pa qV a ≤ qV d = exp  − exp  − d   β   Rp  

(

( )

)

(

)

⇒ qV d R p = α − β ⋅ ln  − ln 1 − 1 / R p 

     

(3.7)

It is the mean wind velocity V d that corresponds to such a value of qV d that is used as a representative basis for the design of structures. R p

is in general subject to

standardisation, e.g. R p = 50 years, in which case qV d (50 ) ≈ α + 3.9 ⋅ β . The ratio

β / α ≈ 0.2 is frequently encountered in the literature. Since qV d = ρV d2 / 2 , then a change from R p = 50 to another return period is given by

( )

V d R p / V d ( 50 ) ≈

{1 − ( β / α ) ⋅ ln − ln (1 − 1 / R )} / (1 + 3.9 ⋅ β / α ) p

(3.8)

While the above considerations are concerned with the statistical properties of annual maxima, it should be mentioned that within any short term (10 min.) stationary weather window at high wind velocities it is possible to estimate an extreme value of the velocity fluctuations. For instance, at any chosen characteristic design value V d ( R p ) , the corresponding extreme value may be obtained by a simple linearization and the broad band type of process considerations shown in chapter 2.4. Since the instantaneous velocity pressure q u (t ) =

{

2 2 2 1 1 1 ρ U (t )  = ρ V + u (t )  = ρV 2 1 + 2u (t ) / V + u (t ) / V  2 2 2

} (3.9)

58


at low turbulence and high values of V can be approximated by

q u (t ) ≈

1 ρV 2 2

 u (t )  1 + 2  V  

(3.10)

it is seen that the mean value of q u is q u = qV = ρV 2 2 while the fluctuating part is ρVu (t ) . The standard deviation of the velocity pressure is then σ qu = ρV σ u , where σ u

is the standard deviation of the along wind turbulence component. Thus, an extreme value of qu may be obtained by qu m a x =

1 ρV m2 a x = q u + k p σ qu 2

(3.11)

where k p is a peak factor (see chapter 2.4, Eq. 2.45). The following is then obtained: 1 1 ρV m2 a x = ρV 2 + k p ρV σ u 2 2

3.2

⇒ V m a x = V 1 + 2k p σ u V

(3.12)

Single point statistics of wind turbulence

While we in the previous chapter were dealing with long term statistics of ten minutes mean values, i.e. performing statistics on a data base covering many years of observations of V , we shall now return to short term statistics on the fluctuating flow components u (t ) , v (t ) and w (t ) . It is single point recordings of these variables within a stationary period of T=10 min that provide the source for determination of their time and frequency domain statistical properties. The sampling frequency within this period is in the following assumed to be large, rendering a sufficiently large data base for the extraction of reliable results. As shown in chapter 1.3, at a certain point ( x , y , z ) f , e.g. at z f = 10 m or at a reference point relevant to the structure in question, it is assumed that U (t ) = V + u (t ) , and that the turbulence components u (t ) , v (t ) and w (t ) are stationary and have zero mean values. For the along wind u

60


(

)

 1 / ln z f / z 0 wh en z f > z m in Iu zf ≈  1 / ln ( z m in / z 0 ) wh en z f ≤ z m in

( )

(3.15)

where z 0 and z m in are defined in Eq. 3.1. Under isotropic conditions (e.g. high above the ground) I u ≈ I v ≈ I w . In homogeneous terrain up to a height of about 200 m and not unduly close to the ground  I v  3 / 4  I  ≈   ⋅ Iu  w  1 / 2 

(3.16)

The auto covariance functions and corresponding auto covariance coefficients (see chapter 2.2) are defined by  Covu (τ )   E u (t ) ⋅ u (t + τ )    1     Covv (τ )  =  E v (t ) ⋅ v (t + τ )   = T Covw (τ )   E w (t ) ⋅ w (t + τ )        

ρn (τ ) =

Cov n (τ )

σ n2

where

 u (t ) ⋅ u (t + τ )    ∫  v (t ) ⋅ v (t + τ )  d t 0 w t ⋅ w t + τ  )  ( ) (

T

n = u ,v ,w

(3.17)

(3.18)

where τ is an arbitrary time lag that theoretically can take any value within ±T . At τ = 0 Eq. 3.17 becomes identical to 3.13, and thus

ρn (τ = 0 ) = 1

where

n = u ,v ,w

(3.19)

At increasing values of τ the auto covariance of the turbulence components diminish, and at large values of τ they asymptotically approach zero, i.e. lim ρn (τ ) = 0

where

n = u ,v ,w

(3.20)

Cov n (τ ) = Cov n ( −τ ) where

n = u ,v ,w

(3.21)

τ →∞

As shown in Eq. 2.19,

implying that also ρn (τ ) is symmetric. A principal variation of the covariance coefficient for the along wind turbulence component is shown in Fig. 3.4. The time scale

3.2 SINGLE POINT STATISTICS OF WIND TURBULENCE

61

∞

T n = ∫ ρn (τ ) dτ

where

n = u ,v ,w

(3.22)

0

may be interpreted as the average duration of a u , v or w wind gust. Although the covariance coefficient in many practical cases may become negative at large values of τ it is a usual approximation to adopt

ρn (τ ) = exp ( −τ / T n )

where

n = u ,v ,w

(3.23)

In homogeneous terrain, at heights below 100 m, T u is usually in the range between 5 and 20 s, while T v and T w are in the ranges 2 – 5 and 0 – 2 s.

Fig. 3.4

Auto covariance coefficient for the along–wind turbulence component

Adopting Taylor’s hypothesis that turbulence convection in the main flow direction takes place with the mean wind velocity (i.e. that flow disturbances travel with the average velocity V ), then the average length scales of u , v and w in the x f direction are given by xf

∞

L n = V ⋅ T n = V ⋅ ∫ ρn (τ ) dτ

where

n = u ,v ,w

(3.24)

0

These turbulence length scales may be interpreted as the average eddy size of the u , v and w components in the direction of the main flow. While auto covariance functions (or coefficients) represent the time domain properties of the turbulence components, it is the spectral densities that describe their frequency domain properties. In the literature many different expressions have been suggested to fit data from a variety of full scale recordings. The following non–dimensional expression proposed by Kaimal et. al. [10] is often encountered in the literature:


62 f ⋅ S n {f }

σ n2

=

(

A n ⋅ fˆn

1 + 1.5 ⋅ A n ⋅ fˆn

x

and where fˆn = f ⋅ f L n / V , and

xf

)

5 /3

where

n = u ,v ,w

(3.25)

L n is the integral length scale of the relevant

turbulence component, as defined in Eq. 3.24 above.

Fig. 3.5

Kaimal auto spectra of turbulence components

Unless full scale recordings indicate otherwise, the following values of the parameter A n may be adopted: A u = 6.8 , A v = A w = 9.4 . With these parameters, Eq. 3.25 has been plotted in Fig. 3.5. Alternatively, the von Kármán [11] spectra

3.3 THE SPATIAL PROPERTIES OF WIND TURBULENCE

f ⋅ S u {f }

=

4 ⋅ fû

( ) ˆ ˆ f ⋅ S { f } 4 f ⋅ (1 + 755.2 ⋅ f ) = σ (1 + 283.2 ⋅ fˆ ) σ u2

1 + 70.8 ⋅ fû2

2 n

(3.26)

5/6

2 n

n

n 2 n

63

11 / 6

, n = v ,w

have the advantage that they contain only the length scales

xf

(3.27)

L n that require fitting to

the relevant data.

3.3

The spatial properties of wind turbulence

The spatial properties of wind turbulence are obtained from simultaneous two point recordings of the u , v and w components. It is taken for granted that the flow is homogeneous in space as well as stationary in time. Defining two vectors  u ( s,t )    υa =  v ( s , t )  w ( s , t )   

and

 u ( s + ∆s , t + τ )    υb =  v ( s + ∆s , t + τ )  w ( s + ∆s , t + τ )   

(3.28)

where s = x f , y f or z f , τ is a time lag that theoretically can take any value within ±T and ∆s is an arbitrary separation (between the two recordings) in the x f , y f or z f

directions. Thus, the following three by three covariance matrix may be defined

 Covu u Cov ( ∆s ,τ ) =  Covvu Covw u

Covu v Covvv Covw v

Covu w  1 Covvw  = E υa ⋅ υTb  = T Covw w 

T

∫ (υa ⋅ υb ) d t T

(3.29)

0

where all the relevant covariance functions

Covm n ( ∆s ,τ )

m , n = u , v , w  ∆s = ∆x f , ∆y f , ∆z f

may contain separation in an arbitrary direction s = x f , y f or z f .

(3.30)

3.3 THE SPATIAL PROPERTIES OF WIND TURBULENCE

S n n ( ∆s , ω ) =

1 2π

+∞

∫ Covn n ( ∆s ,τ ) ⋅ e

−iωτ

dτ

−∞

n = u , v , w  ∆s = ∆x f , ∆y f , ∆z f

67 (3.37)

but in wind engineering the frequency f (in Hz) is usually preferred, and then the double sided cross spectra are defined by (see Eqs. 2.68 and 2.75) S n n ( ∆s , f ) =

+∞

∫ Covn n ( ∆s ,τ ) ⋅ e

−2 π f τ

dτ

−∞

n = u , v , w  ∆s = ∆x f , ∆y f , ∆z f

(3.38)

The cross spectra are usually defined by the single point spectra, S n ( f ) , the coherence function, Coh n n ( ∆s , f ) and the phase spectra, ϕn n ( ∆s , f ) , as shown in Eq. 2.87, i.e.

S n n ( ∆s , f ) = S n ( f ) ⋅ Coh n n ( ∆s , f ) ⋅ exp [iϕ ]

n = u , v , w  ∆s = ∆x f , ∆y f , ∆z f

(3.39)

Since the wind field is usually assumed homogeneous and perpendicular to the span of the (line-like) structure, phase spectra may be neglected. It should however be acknowledged that in structural response calculations spatial averaging takes place along the span of the structure (see chapters 6.4 and 6.5), and then all imaginary parts cancel out and only a double set of real parts remain. Taking it for granted that the single point spectrum S n ( f ) is known, it is then rather the normalised co-spectrum ˆ Co n n ( ∆s , f ) =

Re S n n ( ∆s , f ) 

n = u , v , w  ∆s = ∆x f , ∆y f , ∆z f

Sn (f )

(3.40)

that is necessary to give special attention to in wind engineering. Some general expressions occur in the literature. For a first approximation and under homogeneous conditions, the following may be adopted

ˆ Co nn

 f ⋅ ∆s ( ∆s , f ) = exp  −cn s ⋅ V zf 

where:

( )

cn s

   

n = u , v , w  s = x f , y f , z f  ∆s = ∆x f , ∆y f , ∆z f

c = cu z f ≈ 9  uyf  = cvy f = cvz f = cw y f ≈ 6  cw z f ≈ 3

(3.41)

(3.42)

70

4 BASIC THEORY OF STOCHASTIC DYNAMIC RESPONSE CALCULATIONS

each containing three components (horizontal, vertical and torsion), i.e.:

r ( x ) = ry q ( x ) = q y

rz qz

rθ 

T

qθ 

r ( x , t ) = ry T

q ( x , t ) = q y

rz qz

rθ 

T

qθ 

(4.2) T

(4.3)

In the following the mean values of the response are considered trivial, and the entire focus is on the calculation of the variances of the fluctuating displacement components. The solution will be based on a modal frequency domain approach. Thus, it is assumed that a sufficiently accurate eigen–value solution is available, and that it contains the necessary number of eigen-frequencies and corresponding eigen-modes. That they are orthogonal goes without saying. Scaling of mode shapes is optional, but consistency is required such that the relative difference between cross sectional displacement and rotation components is maintained. It is taken for granted that it has been obtained in vacuum or in still air conditions. Such a solution has usually been obtained from some finite element formulation, and for line-like beam type of elements the eigen-modes will then occur as vectors containing six components in each element node, three displacements and three rotations. In the development of the theory below the number of eigen–value components is reduced, focusing on the degrees of freedom associated with ry , rz and rθ . Thus, the mode shape components associated with an arbitrary mode is the displacements φ y , φz and the cross sectional rotation φθ . It should be noted that they are formally treated as continuous functions, and therefore the two other rotation components may be retrieved from the first derivatives of φ y and φz . It is then only the x–axis displacement (i.e. the axial component in the direction of the main span) that is entirely discarded, but this is not considered important since it is not directly associated with any flow induced load effects and it is usually small as compared to the other components.

Example 4.1: Given a simply supported beam with a single symmetric channel type of cross section as shown in Fig. 4.2. The system contains three displacement components: y(x,t), z(x,t) and θ(x,t), all referred to the shear centre, which in this case does not coincide with the centroid. Disregarding any external loading and damping contributions, the differential equilibrium conditions are given by (see Timoshenco, Young & Weaver [1], chapter 5.21):

4.1 MODAL ANALYSIS AND DYNAMIC EQUILIBRIUM EQUATIONS

EI z EI y

∂4 y 4

∂x

∂4 z ∂x

4

∂ 2θ

+my +mz

∂t 2

( y − e ⋅θ ) = 0

∂2 z

=0

∂t 2 ∂ 4θ

2

4

+my

∂2 y 2

⋅ e − mθ

∂ 2θ

=0 ∂x ∂x ∂t ∂t 2 where EIz and EIy are cross sectional stiffness with respect to bending in the y and z directions, GIt and EIw are the corresponding torsion stiffness associated with St.Venant’s torsion and warping, my and mz are translational mass (per unit length), mθ is rotational mass (with respect to the shear centre) and e is the vertical distance from the shear centre to the centroid. Obviously my=mz (for GI t

− EIw

∂2

71

simplicity they are both set equal to m) and m θ = ρ I p + m e2

Simply supported beam with channel type of cross section

Fig. 4.2

These equations are satisfied over the entire span for the following displacement functions T a =  a  y ( x , t ) a z aθ  y     where  z ( x ,t ) = a ⋅ f ( x ,t ) nπ x  f ( x , t ) = sin ⋅ exp ( iωt ) θ ( x , t )    L  and n = 1, 2,......., N . Introducing this into the differential equations above, the following eigen-

(

)

value type of problem is obtained: K − ω 2M ⋅ a = 0 , where:  n π   L  K =     

4

  EI z 

0 4

0

 nπ   L  EI y  

0

0

     0   2 n 2π 2 E I w    nπ      L   GI t + L2     

0

0  m  and M =  0 m  −m ⋅ e 0

−m ⋅ e   0  m θ 

(Under more general conditions a Galerkin type of solution procedure is applicable.) There are two independent eigen–value solutions to this problem. First, there is one that only involves z(x,t) displacements, defined by


72

 n π 4  2   E I y − ω m  ⋅ az = 0 L    which will render n eigen–values and corresponding eigen–vectors 0  nπ x   φ1n ( x ) = sin ω1n = ( n π ) and 1 L   m L4 0  The second solution involves a combined motion of y(x,t) and θ(x,t) displacements. It is defined by 2

 n π   L    

EI y

4

 2  EIz −ω m 

ω 2m e

   a y    =0 2 2 2 n π EIw   nπ   2   aθ   − ω mθ   L   GI t + 2  L     

ω 2m e

and it will render two different eigen–values and corresponding eigen–vectors:

ω2n

2  1 + ωˆ +  1 − ωˆ  + e2 Kˆ  2   2   

    

ω3n

  Kθ = 2  m θ − e m

2  1 + ωˆ −  1 − ωˆ  + e2 Kˆ    2  2  

     

4

 nπ  where: K z =   EI z ,  L  aˆθ2 =

φ2n

 1   nπ x   0  ( x ) = sin L   aˆ θ2 

φ3 n

 1   nπ x   0  ( x ) = sin L   aˆ θ3 

1/2

  Kθ = 2  m θ − e m

1/2

2 2   nπ    nπ  Kθ =  EIw  , GI t +     L    L  

1  ωˆ − 1  ωˆ − 1 2 ˆ +  −e K  e  2  2

   

aˆθ3 =

K Kˆ = z , Kθ

ωˆ =

1  ωˆ − 1  ωˆ − 1 2 ˆ −  −e K  e  2  2

K z mθ m Kθ

   

It may be of some interest to develop the modal mass associated with these mode shapes. The cross sectional mass matrix is given by M0 = d iag [m m m θ ] , and thus L

L

nπ x M 1n = ∫ φ1Tn M0 φ1n d x = m ∫ sin 2 dx = m L / 2 L 0 0 L

(

M 2n = ∫ φT2n M0 φ2n d x = m + aˆθ22 m θ 0

L

L

) ∫ sin 0

2

nπ x d x = m + aˆθ22 m θ L / 2 L

M 3n = ∫ φT3n M0 φ3n d x =M 2n 0

(

)


73

In this case mode shapes have been normalised with the displacement component, and therefore the rotation component in the mode shape vector has the unit m-1 while the modal mass has the unit kg.

In a general structural eigen–value problem

(K − ω M ) ⋅ Φ = 0 2

(4.4)

the modes are usually defined M-orthonormal, i.e. such that

ΦT ⋅ M ⋅ Φ = I

(4.5)

where I is the (diagonal) identity matrix. It should be acknowledged that prior to any scaling of the modes their components have units meters or radians, and that after any scaling has taken place relative units must be maintained (as shown in the example above). It should also be acknowledged that the importance of a rotational component should not be underestimated although the absolute values of the elements in its eigenmode vector are small as compared to those of the corresponding displacement components. The typical mode shapes from such a general solution is illustrated in Fig. 4.3, where φ the shape vector for each mode has been reduced to the three relevant φ y , φz and θ components. In the mathematical development of a frequency domain response calculation theory that follows below, the cross sectional displacement and load components are as mentioned above formally taken as continuous function. The motivation behind this choice is mainly convenience, but it is also for practical reasons as spatial load integration will most often require mode shape vectors in a considerably finer element mesh than what is considered sufficient for the eigen–value solution from which they have been obtained. After the theory has been developed the return to discrete vectors will be shown wherever this is necessary for a convenient numerical solution. The basic assumption behind a modal approach is that the structural displacements r ( x , t ) may be represented by the sum of the products between natural eigen–modes

φi ( x ) = φ y

φz

φθ 

T

i

(4.6)

and unknown exclusively time dependent functions ηi (t ) , i.e.

r ( x ,t ) =

N m od

∑ i =1

φ y ( x )    φz ( x )  ⋅ ηi (t ) = Φ ( x ) ⋅ η (t ) φ ( x )   θ i

(4.7)


(

= d iag  M   M 0  i   C0 = d iag C i    K 0 = d iag  K i   

)

 M = φT ⋅ M ⋅ φ d x i 0  i L∫ i  Ci = 2 M i ωi ζ i  2  K i = ωi M i 

where

75

(4.10)

The modal load vector in Eq. 4.9 is given by = Q Q tot  1

where: Q itot =

∫

L exp

... Q i

(φ

T i

T ... Q N m od  tot

)

⋅ qtot d x

(4.11) (4.12)

In Eq. 4.10 ωi are the eigen-frequencies and ζ i are the damping ratios, each associated with the corresponding eigen-mode. It is in the following assumed that the structural damping ratios ζ i are known quantities, chosen from experimental experience or an acknowledged code of practice, and that a pertinent mode shape variation has been adopted (e.g. a Rayleigh type of frequency dependency). The three by three mass matrix M0 = d iag m y ( x ) m z ( x ) m θ ( x ) 

(4.13)

contains the cross sectional mass properties associated with the y, z and θ degrees of freedom, all taken with respect to the shear centre. (It may often be more convenient to in Eq. 4.9 directly from the nodal mass lumping used in calculate modal mass matrix M 0 the preceding finite element eigen–value solution and the corresponding eigen-vectors, instead of the formal calculation procedures indicated above, as these already contain all the structural properties that are necessary for such a calculation.) The cross sectional load vector qtot contains the total drag, lift and moment loads per unit length (see Fig. 4.1) including flow induced as well as motion induces loads, i.e. qt ot = q y

qz

T

q θ t ot

(4.14)

The symbolic integration limits L and Lexp indicate integration over the entire structure or and K on the , C over the wind exposed part of the structure. The modal matrices M 0 0 0 left hand side of Eq. 4.9 are all diagonal due to the orthogonal properties of the eigenmodes. However, we shall later see that motion induced parts of the load will be moved to the left hand side of the modal equilibrium equation, thus rendering non-diagonal mass, damping and stiffness matrices for the combined flow and structural system. For educational reasons the development of the necessary theory is divided into three parts,

76


depending on the complexity of the problem. The first part of the presentation is dealing with the situation that the relevant eigen-frequencies are well separated and each mode only contains one component. The next is dealing with the same situation but now with each mode containing all three components. The final presentation is considering the situation that a full multi-mode investigation is required.

4.2

Single mode single component response calculations

In this first section it is assumed that the eigen-frequencies are well spaced out on the frequency axis. Furthermore, the cross sectional shear centre is assumed to coincide (or nearly coincide) with the centroid and there are no other significant sources of mechanical or flow induced coupling between the three displacement components. These assumptions imply that coupling between modes may be ignored, and that each mode shape only contains one component, i.e. any of the N m od mode shapes is purely horizontal, vertical or torsion. The response covariance between modes will then be zero, and thus, the variance of the total dynamic horizontal, vertical or torsion displacement response can be obtained as the sum of contributions from each mode, i.e. the variance of a displacement component is the sum of all variance contributions from excited modes containing displacement components exclusively in the y, z or θ direction (see Eq. 2.27). E.g. σ y2 is the sum of all variances associated with the relevant number of modes containing only horizontal displacements, and so on. Thus,

  σ i2y   ∑ σ y2   i y      2 2 σ z  =  ∑ σ i   2   iz z  σ θ   2     ∑ σ iθ   iθ 

(4.15)

4.2 SINGLE MODE SINGLE COMPONENT RESPONSE CALCULATIONS

77

Given an arbitrary horizontal, vertical or torsion mode shape φi ( x ) with eigen– frequency ωi and damping ratio ζ i , the time domain displacement response contribution of this mode is simply

ri ( x , t ) = φi ( x ) ⋅ ηi (t )

(4.16)

As mentioned above, it is assumed that the corresponding instantaneous cross sectional load contains the sum of flow induced and motion induced contributions. Thus, the total load per unit length (horizontal, vertical or torsion) is given by qtot = q ( x , t ) + q ae ( x , t , ri , ri , ri )

(4.17)

where q ( x , t ) is the flow induced part and q ae ( x , t , ri , ri , ri ) is the additional load associated with interaction between flow and structural motion. The modal time domain equilibrium equation for mode number i is then given by M i ⋅ ηi (t ) + C i ⋅ ηi (t ) + K i ⋅ ηi (t ) = Q i (t ) + Q aei (t ,ηi ,ηi ,ηi )

(4.18)

where

 φi2 m d x    M i   L∫      C i  = 2 M i ωi ζ i      2  K i  ωi M i      Q i (t )  q    = ∫ φi ⋅   d x Q aei (t ,η ,η ,η)  L exp q ae 

          

(4.19)

L exp is the flow exposed part of the structure and Q aei (t ,ηi ,ηi ,ηi ) is the modal motion induced load. It should be noted that structural mass m ( x ) in the equation above will either be translational or rotational (with respect to the shear centre), depending on

78


whether the mode shape involves displacements in the y or z directions or if it involves pure torsion. Transition into the frequency domain is obtained by taking the Fourier transform on either side of Eq. 4.18. Thus,

( − M ω i

2

)

+ C i iω + K i ⋅ aηi (ω ) = a Q (ω ) + a Qae (ω ,ηi ,ηi ,ηi ) i

(4.20)

i

where aηi , a Q and a Qae are the Fourier amplitudes of ηi (t ) , Q i (t ) and Q aei (t ) , i

i

respectively. (Index i is the mode shape number and the symbol i is the imaginary unit i = −1 .) It is now assumed that the Fourier amplitude of the modal motion induced load contains three cross sectional terms k ae , cae and m ae , proportional to and in phase

with structural displacement, velocity and acceleration, i.e. it is assumed that

(

)

a Qae = − M aei ω 2 + C aei iω + K aei ⋅ aηi i

(4.21)

where

 M ae  i   2  C ae  = ∫ φi i   L exp  K ae  i  

m ae   c dx  ae   k ae 

(4.22)

and where k ae , cae and m ae are known constants. Introducing a Qae from Eq. 4.21 (as i

well as C i = 2 M i ωi ζ i and K i = ωi2 M i from Eq. 4.10) into Eq. 4.20, gathering all

motion dependent terms on the left hand side and dividing throughout the equation by K , the following is obtained i

aηi (ω ) =

Hˆ i (ω ) ⋅ a Q (ω ) i K

(4.23)

i

where   K ae M aei Hˆ i (ω ) = 1 − 2 i − 1 −  ωi M i  M i  

   ω 2 C aei  ⋅   + 2i  ζ i −  ω  2ωi M i   i 

is the non-dimensional modal frequency-response-function. Introducing µ = M M , κ = K ω 2 M and ζ aei

aei

i

aei

aei

i

i

aei

 ω ⋅   ωi   

(

)

(

)

(4.24)

( 2ωi M i ) , then

= C aei

2  ω ω ˆ  H i (ω ) = 1 − κ aei − 1 − µaei ⋅   + 2i ζ i − ζ aei ⋅  ωi    ωi   

−1

−1

(4.25)

4.2 SINGLE MODE SINGLE COMPONENT RESPONSE CALCULATIONS

79

The single–sided spectrum of ηi (t ) is given by 2

S ηi

Hˆ i (ω ) 1 1 ⋅ aη*i ⋅ aηi = ⋅ lim ⋅ a Q* ⋅ a Q (ω ) = Tlim 2 i i T →∞ π T →∞ π T Ki

(

(

)

⇒ S ηi (ω ) =

Hˆ i (ω )

)

(4.26)

2

⋅ S Q (ω )

K i2

(4.27)

i

where it has been introduced that the single-sided spectrum of the modal loading is defined by 1 S Q (ω ) = lim ⋅ a Q* ⋅ a Q (4.28) i i i T →∞ π T

(

)

This will render the displacement response at a position where φi ( x ) = 1 . The response

at an arbitrary position x r (e.g. where φ has its maximum) may simply be obtained by recognizing that due to linearity the Fourier amplitude at x r is given by a r (ω ) = φi ( x r ) ⋅ aη (ω ) i

(4.29)

i

and thus, the response spectrum for the displacement response at x = x r is given by S r ( x r ,ω ) =

φi2 ( x r ) K i2

i

2

⋅ Hˆ i (ω ) ⋅ S Q (ω )

(4.30)

i

In structural engineering it has been customary to split the response calculations into a background and a resonant part as illustrated in Fig. 4.4. The motivation behind this is that static and quasi-static load effects are more accurately determined directly from time invariant equilibrium conditions. This is particularly important for the determination of cross sectional force resultants (or stresses), as shown in chapter 7. The variance of the displacement response in Eq. 4.30 split into a background and a resonant part is given by

σ r2i ( x r ) = ≈

2 φi2 ( x r ) ∞ ˆ ∫ H i (ω ) ⋅ S Q (ω ) d ω 2

K i

i

0

∞ ∞  2 2 φi2 ( x r )  ˆ ˆ (ω ) d ω  ⋅ ⋅ + ⋅ 0 H S ω d ω S ω H  ( ) ( ) ( ) i i i ∫ ∫ Q Q 2

K i



0

i

i

0



    (4.31)   

82


is the motion induced part. The time domain modal equilibrium condition given in Eq. 4.18 still holds, but for the expressions for M , C and K it is necessary to turn to Eq.

(

)

4.10, i.e. M i = ∫ φTi ⋅ M0 ⋅ φi d x , C i = 2 M iωiζ i and K i = ωi2 M i while the modal flow L

induced part of the load is given by (see Eq. 4.12)

Q i (t ) =

∫

φTi ( x ) ⋅ q ( x , t ) d x

(4.39)

L exp

The Fourier transform in Eq. 4.20 as well as the assumption regarding a Qae in Eq. 4.21 i

are also still valid, but again, modal motion induced mass, damping and stiffness are now given by

φTi ⋅ Mae ⋅ φi   M ae   T     C ae  = ∫  φi ⋅ Cae ⋅ φi  d x    L exp  T  K ae i  φi ⋅ K ae ⋅ φi 

(4.40)

where Mae , Cae and K ae are three by three coefficient matrices associated with the motion induced part of the loading. To justify a mode by mode approach it is necessary to avoid the introduction of any motion induced coupling between modes, and therefore Mae , Cae and K ae must in this particular case be diagonal, i.e. M ae = d iag m y

mz

C ae = d iag c y

cz

K ae = d iag  k y

kz

m θ  cθ 

ae

kθ 

ae

ae

     

(4.41)

Thus, altogether nine frequency domain motion dependent coefficients are required. In wind engineering Mae is most often negligible. Modally we are still dealing with a single-degree-of-freedom system, and thus, Eqs. 4.24 – 4.27 are valid. Linearity implies that the Fourier amplitudes of the displacement components at an arbitrary position x r are given by  a ry  φ y ( x r )      ar ( x r , ω ) =  a rz  = φz ( x r )  ⋅ aη (ω ) = φr ( x r ) ⋅ aη (ω ) i i i      a rθ  φθ ( x r )  i The cross spectral density matrix of the three components is then

(4.42)

4.3 SINGLE MODE THREE COMPONENT RESPONSE CALCULATIONS

Si ( x r , ω ) = lim

1

T →∞ π T

( ari ⋅ ari ) *

(

T

1   φri ⋅ aηi T →∞ π T 

= lim

83

) ⋅ (φri ⋅ aηi ) *

T

(

)

1  aη* ⋅ aη ⋅ φTr (4.43)  = φri ⋅ lim i i i →∞ T T π 

from which the following is obtained $ %   = φ x ⋅ S ω ⋅ φT S n m ( x r ,ω ) Si ( x r , ω ) =  ri ( r ) ηi ( ) ri  $  % i where:

(

n  1 aη* ⋅ aη  = ry , rz , rθ and S ηi (ω ) = lim i i T →∞ π T m

)

(4.44)

is given in Eqs. 4.26 and

4.27, i.e.

Si ( x r , ω ) = φr ( x r ) ⋅ i

Hˆ i (ω ) K i2

2

⋅ S Q (ω ) ⋅ φTr i

i

(4.45)

The response covariance matrix is obtained by the frequency domain integration of Si ( x r , ω ) , and thus  σ 2 ( x ) Cov  ry rz ( x r ) Cov ry rθ ( x r )  ry ry r  ∞   2 ∫ Si ( x r , ω ) d ω = Covrz ry ( x r ) σ rz rz ( x r ) Covrz rθ ( x r )  0   Cov rθ ry ( x r ) Cov rθ rz ( x r ) σ r2θ rθ ( x r )   i

(4.46)

However, the three components of each mode shape are fully correlated and therefore all cross-covariance coefficients that may be extracted from Eq. 4.46 are equal to unity. Thus, it is only the terms on the diagonal of Eq, 4.46 that are of any interest, and then the calculations simplify into φ y2 ( x r )  S ry ry  2   Hˆ i (ω )   Si ( x r , ω ) =  S rz rz  = φz2 ( x r )  ⋅ ⋅ S Q (ω ) i  2  K i2   S rθ rθ i φθ ( x r )  i

(4.47)

84

and


σ r2 r   yy ∞ Vari ( x r ) = σ r2z rz  = ∫ Si ( x r , ω ) d ω   0 σ 2   rθ rθ i

(4.48)

The total response may be obtained by adding up variance contributions from all modes, i.e. σ r2 r ( x r )   yy  N m od Var ( x r ) = σ r2z rz ( x r )  = ∑ Vari (4.49)   i =1 σ 2   rθ rθ ( x r ) 

4.4

General multi-mode response calculations

In the final section of this chapter it is assumed that a full multi-mode approach is required. The basic assumptions from chapter 4.1 are that

r ( x , t ) = Φ ( x ) ⋅ η (t )

where

      

(4.50) T

r ( x , t ) = ry rz rθ  Φ ( x ) = φ1 ... φi ...φ N m od  η ( x ) = η1 ... ηi T

...ηN m od 

(4.51)

T

φi ( x ) = φ y φz φθ  and where N m od is the number of modes chosen to be i included in the calculations. Still adopting the assumptions regarding motion induced load effects as presented in chapter 4.2, the cross sectional load is qtot = q ( x , t ) + qae ( x , t , r , r, r) (4.52)

4.4 GENERAL MULTI-MODE RESPONSE CALCULATIONS

    

where

q ( x , t ) = q y

qθ 

qz

qae ( x , t , r , r, r ) = q y

85

T

qθ 

qz

(4.53)

T ae

Thus, the time domain modal equilibrium equation is given by (see also Eq. 4.9) ⋅ η (t ) + K ⋅ η (t ) = Q (t ) + Q (t ,η ,η ,η) ⋅η (t ) + C M 0 0 0

(4.54)

and K are N , C where M m od by N m od diagonal matrices defined in Eq. 4.10, and 0 0 0 the modal N m od by one flow induced load vector is given by T (t ) = Q ... Q ... Q  Q i N m od   1 Q i = ∫ φTi ⋅ q d x

where

L exp

(

(4.55)

)

(4.56)

Taking the Fourier transform on either side of Eq. 4.54

( −M ω 0

2

)

iω + K ⋅ a (ω ) = a (ω ) + a +C 0 0 η Q Q     

where:

aη = aη1 .... aηi

ae

....aηN 

aQ = a Q .... a Q i  1

....a Q

N

(ω ,η ,η,η)

(4.57)

T

 

(4.58)

T

Since the assumption of a modal frequency domain motion induced load proportional to and in phase with structural displacement, velocity and acceleration is adopted, then aQ is given by ae

(

)

iω + K ω2 + C aQae = −M ae ae ae ⋅ aη

(4.59)

and K are N , C where M m od by N m od matrices ae ae ae

M ae

%  = M aeij  $

$    %

C ae

%  C aeij =  $

$    %

whose elements on row i column j are given by

K ae

%  K aeij =  $

$    %

(4.60)

86


 M  φTi ⋅ Mae ⋅ φ j   aeij     C =  φTi ⋅ Cae ⋅ φ j  d x ae ∫  ij      L exp  φT ⋅ K ⋅ φ  K i ae j ae    ij  

(4.61)

where Mae , Cae and K ae are three by three motion dependent cross sectional load coefficient matrices

% Mae =  m nm $ and where:

$   %

% cn m Cae =  $

$   %

% kn m K ae =  $

$   %

(4.62)

n   = y , z ,θ . m

Example 4.2: The modal quantities given in Eq. 4.61 may be obtained from a fully expanded vector format given by:

M aeij = (φ yTi m yy φ y j + φzTi m zy φ y j + φθTi m θ y φ y j + φ yTi m yz φz j + φzTi m zz φz j + φθTi m θ z φz j +φ yTi m yθ φθ j + φzTi m zθ φθ j + φθTi m θθ φθ j ) ⋅ ∆x C aeij = (φ yTi c yy φ y j + φzTi czy φ y j + φθTi cθ y φ y j + φ yTi c yz φz j + φzTi czz φz j + φθTi cθ z φz j +φ yTi c yθ φθ j + φzTi czθ φθ j + φθTi cθθ φθ j ) ⋅ ∆x K aeij = (φ yTi k yy φ y j + φzTi k zy φ y j + φθTi kθ y φ y j + φ yTi k yz φz j + φzTi k zz φz j + φθTi kθ z φz j +φ yTi k yθ φθ j + φzTi k zθ φθ j + φθTi kθθ φθ j ) ⋅ ∆x where ∆x is the spanwise mesh separation (above assumed constant). If the coefficients vary along the span, their numerical values need to be given on the diagonal of an N by N matrix, where N is the number of nodes.

Thus, for a full description of the motion induced load effects altogether twenty–seven motion dependent load coefficients are required. First Eqs. 4.59 is introduced into 4.57 and all terms associated with structural motion are gathered on the left hand side


(

(

)

)

(

87

)

2 −M  − M 0 ae ω + C0 − Cae iω + K 0 − K ae  ⋅ aη (ω ) = aQ ( ω ) 

(4.63)

and then the result is pre-multiplied with K 0−1 , recalling that

   = d iag 2 M ω ζ   C 0 i i i  

K 0 = d iag ωi2 M i 

(4.64)

It is convenient to introduce a reduced modal load vector

a ˆ (ω ) = K 0−1 Q

  ⋅ aQ (ω ) = "   

where aq ( x , ω ) =  a q y 

a qz

φTi ( x ) ⋅ aq ( x , ω ) d x

∫

L exp

ωi2 M i

  "   

T

(4.65)

T

a qθ  . The following is then obtained 

ˆ (ω ) ⋅ a (ω ) aη (ω ) = H η Qˆ

(4.66)

where −1

       iω   ω 2 +  d iag  2ζ i  − K −1 C ˆ (ω ) = I − K −1K −  d iag  1  − K −1 M H    ae ae ae  η 0 0 0 2     ωi      ωi     (4.67) is the non-dimensional frequency-response-matrix, and I is the identity matrix ( N m od by N m od ). It is convenient to define the following N m od by N m od matrices

(

µae = d iag ωi2  ⋅ K 0−1 ⋅ M ae

)

κ ae = K 0−1 ⋅ K ae

(

)

      

(4.68)

1 ⋅ d iag [ωi ] ⋅ K 0−1 ⋅ C ae 2 as well as introducing ζ = d iag [ζ i ] . The non-dimensional frequency-response-matrix ζ ae =

is then given by

88


2     1  1    ˆ Hη (ω ) = I − κ ae −  ω ⋅ d iag    ⋅ ( I − µae ) + 2iω ⋅ d iag   ⋅ ( ζ − ζ ae )     ωi    ωi    

−1

(4.69) By combination of Eqs. 4.60, 4.61 and 4.64, then the content of

µae

%  µaeij =  $

% $   κ aeij  κ ae =    % $

are given by

µaeij =

κ aeij =

ζ aeij =

∫

M aeij

=

M i

L exp

∫

K aeij

L exp

=

ωi2 M i

2ωi M i

ζ ae

(φ

T i Mae φ j

(φ

T i K ae φ j

=

L exp

(φ

T i

%  = ζ aeij  $

$   (4.70)  %

)dx (4.71)

M i

ωi2 M i

∫

C aeij

$    %

)dx (4.72)

)

Cae φ j d x (4.73)

2ωi M i

Returning to Eq. 4.66, the response spectral density matrix ( N m od by N m od and containing single-sided spectra) is obtained from the basic definition of spectra as expressed from the Fourier amplitudes, and thus, the following development applies:

(

(

)

) (

* 1 1  ˆ ˆ a aη* ⋅ aηT = lim Hη aQˆ ⋅ H η Qˆ  T →∞ π T T →∞ π T  ˆ * ⋅  lim 1 a* ⋅ aT  ⋅ H ˆT = H ˆ * ⋅ S ⋅H ˆT =H ˆ ˆ  ˆ η  η η η Q Q Q T →∞ πT  

Sη (ω ) = lim

(

)

where SQˆ is an Nmod by Nmod normalised modal load matrix

)

T

  

(4.74)


(

1 aQ*ˆ ⋅ aTQˆ T →∞ π T

SQˆ (ω ) = lim

)

  a *ˆ   Q1  #  1   a * = lim Qˆ i T →∞ π T    #  *   a Qˆ   N m od

     ⋅ a ˆ   Q1    

" a Qˆ

% $    S Qˆ Qˆ (ω ) ⇒ SQˆ (ω ) =  i j   $ % whose elements on row i column j are given by

j

89

" a Qˆ

N m od

          

(4.75)

1  * a ˆ (ω ) ⋅ a Qˆ (ω )   j  Qi T   * T  T   ∫ φi ( x ) a q ( x , ω ) d x ∫  φ j ( x ) a q ( x , ω )  d x  L exp 1  L exp  = lim ⋅   2 2 T →∞ π T ωi M i ωj M j       T   *  T   T   ∫∫ φi ( x1 ) aq ( x1 , ω )  ⋅ φ j ( x 2 ) aq ( x 2 , ω )  d x1 d x 2  1  L exp  = lim   2 2 T →∞ π T ωi M i ⋅ ω j M j       1 T a*q ( x1 , ω ) ⋅ aTq ( x 2 , ω )  ⋅ φ j ( x 2 ) d x1 d x 2 ∫∫ φi ( x1 ) ⋅ Tlim  →∞ π T  L exp = ωi2 M i ⋅ ω 2j M j

S Qˆ Qˆ (ω ) = lim i j

T →∞ π T

(

(

)(

)(

)

)

(4.76)

Thus, the elements of SQˆ (ω ) are given by

⇒ S Qˆ Qˆ (ω ) =

∫∫

φTi ( x1 ) ⋅ Sqq ( ∆x , ω ) ⋅ φ j ( x 2 ) d x1 d x 2

L exp

(ω M ) ⋅ (ω M ) 2 i

i j

i

2 j

(4.77)

j

where ∆x = x1 − x 2 , and where Sqq ( ∆x , ω ) is the spectral density matrix of cross sectional loads, i.e. Sqq ( ∆x , ω ) = lim (1 / π T ) aq* ( x1 , ω ) ⋅ aTq ( x 2 , ω )  T →∞

90


a * ⋅ a  qy qy 1  * aq ⋅ aq y ⇒ Sqq ( ∆x , ω ) = lim T →∞ π T  z  * a ⋅a  qθ q y

a q* y ⋅ a q z a q*z ⋅ a q z a q*θ ⋅ a q z

a q* y ⋅ a qθ  S q q   y y * a q z ⋅ a qθ  =  S q z q y   S * a qθ ⋅ a qθ   qθ q y 

Extracting from the mode shape matrix Φ = [φ1

... φi

... φ N

S q y qz S qz qz S qθ q z

]

S q y qθ   S q z qθ   S qθ qθ   (4.78)

(see Eq. 4.8) a

three by N m od matrix associated with a chosen span-wise position x r Φr ( x r ) = φ1 ( x r ) .... φi ( x r ) .... φ N ( x r )   φ y ( x r )  φ y ( x r )  φ y ( x r )        =  φz ( x r )  .... φz ( x r )  .... φz ( x r )    φ ( x )  φ ( x )   φθ ( x r ) 1  θ r i  θ r N

    

(4.79)

then the three by three cross spectral density matrix of the unknown modal displacements ry , rz and rθ at x = x r S  ry ry Srr ( x r , ω ) =  S rz ry  S r r  θy

S ry rz S rz rz S rθ rz

S ry rθ   S rz rθ   S rθ rθ  

(4.80)

is given by

Srr ( x r , ω ) = Φr ( x r ) ⋅ Sη (ω ) ⋅ ΦTr ( x r )

(4.81)

where Sη (ω ) is given in Eq. 4.74, i.e.:

ˆ * ω ⋅ S ω ⋅H ˆ T ω  ⋅ ΦT x Srr ( x r , ω ) = Φr ( x r ) ⋅ H  η ( ) Qˆ ( ) η ( )  r ( r )

(4.82)

This equation is applicable to any linear load on a line–like structure. If all mechanical properties of the structure are known, then an eigen–value analysis will provide the basic ˆ and Φ . What then remains is the set-up of S and the motion induced input to H η

r

ˆ η . This is shown in chapters 5 and 6. contributions to H

Qˆ

Chapter 5 WIND AND MOTION INDUCED LOADS

5.1

The buffeting theory

The buffeting wind load on structures includes the part of the total load that may be ascribed to the velocity fluctuations in the oncoming flow, U ( x f , y f , z f , t ) = V ( x f , y f , z f ) + u ( x f , y f , z f , t ) , v ( x f , z f , t ) and w ( x f , z f , t ) , as well as any motion induced contributions. The theory presented below was first developed by A.G. Davenport [13, 14]. In the following it is a line like horizontal bridge type of structure that is considered. It is taken for granted that its z f –position in the flow prior to any loading is constant along the entire span, that the wind field is stationary and homogeneous and that the main flow direction is perpendicular to the span-wise x -axis of the structure, in which case x f is constant and y f may be exchanged by x . It is then only the velocity fluctuations in the along wind and the across wind vertical directions expressed in structural axis that are of interest, i.e. the components U ( x , t ) = V + u ( x , t ) and w ( x , t ) . The theory may readily be applied to a vertical (tower) type of structure, in which case any z f -variation needs to be included and the w component must be replaced by the v component (but maintaining all other notations shown in Fig. 5.1 below). The basic assumptions behind the buffeting theory are that the load may be calculated from the instantaneous velocity pressure and the appropriate load coefficients that have been obtained from static tests, and that linearization of any fluctuating parts will render results with sufficient accuracy. Thus, the load may be calculated from an interpretation of the instantaneous relative velocity vector and the corresponding flow incidence dependent drag, lift and moment coefficients that are usually applied to calculate mean static load effects. It is taken for granted that structural displacements and cross sectional rotations are small. Furthermore, it is a requirement for linearization of load components in is illustrated that u ( x , t ) and w ( x , t ) are small as compared to V. The situation Fig. 5.1.

5.1 THE BUFFETING THEORY

 q D ( x ,t )    1 2  q L ( x , t )  = 2 ρV rel q M ( x , t )   

93

 D ⋅ C (α )  D   ⋅  B ⋅ C L (α )   2   B ⋅ C M (α ) 

(5.1)

where V rel is the instantaneous relative wind velocity and α is the corresponding angle of flow incidence. Transformation into structural axis is given by q y    qtot ( x , t ) =  q z  q   θ

tot

cos β = sin β  0

− sin β cos β 0

0   qD 0  ⋅  q L 1  q M

   

(5.2)

where:

 w − rz  V + u − ry 

β = a r ct a n 

   

(5.3)

The first linearization involves the assumption that the fluctuating flow components u ( x , t ) and w ( x , t ) are small as compared to V, and that structural displacements (as well as cross sectional rotation) are also small. Then cos β ≈ 1 and sin β ≈ t a n

β ≈ β ≈ (w − rz ) / (V + u − ry ) ≈ (w − rz ) / V , and thus

(

2 V rel = V + u − ry

2 ) + (w − rz )2 ≈ V 2 + 2Vu − 2Vry 

α = rθ + rθ + β ≈ rθ + rθ +

w rz − V V

  

(5.4)

The second linearization involves the flow incidence dependent load coefficients. As illustrated in Fig. 5.2, the nonlinear variation of the load coefficient curves is replaced by the following linear approximation  C D ( α )   C D (α )       C L (α )  =  C L ( α )  + α f C M (α )  C M (α )     

 C D′ (α )    ⋅  C L′ (α )  C M   ′ (α ) 

(5.5)

where α and α f are the mean value and the fluctuating part of the angle of incidence, ′ are the slopes of the load coefficient curves at α . and where C D′ , C L′ and C M

5.1 THE BUFFETING THEORY

95

q y ( x )  q y ( x , t )      qtot ( x , t ) =  q z ( x )  +  q z ( x , t )  = q + Bq ⋅ v + Cae ⋅ r + K ae ⋅ r q ( x )   q ( x , t )   θ   θ 

(5.8)

where

v ( x , t ) = [u w ]

T

r ( x , t ) = ry

q y    ρV 2 B q ( x ) = q z  = 2 q   θ 2 ( D / B ) C D  ρVB  Bq ( x ) = 2C L 2   2B CM  2 ( D / B ) C D  ρVB  Cae ( x ) = − 2C L 2   2B CM 

K ae ( x ) =

2

rθ 

T

(5.10)

( D / B ) C D    ρV 2 B ˆ C L   = 2 ⋅ bq  BC  M  

( ( D / B ) C D′ − C L ) ˆ ⋅B (C L′ + ( D / B ) C D ) = ρVB q 2 ( ( D / B ) C D′ − C L ) (C L′ + ( D / B ) C D )

0 0 0 0 

(5.11)

(5.12)

 

′ B CM

0 0

ρV 2 B 

rz

(5.9)

′ B CM

0  0  0 

(5.13)

( D / B ) C D′  C L′ ′ B CM

   

(5.14)

It is seen that the total load vector comprises a time invariant mean (static) part q y    ρV 2 B ˆ ⋅ bq q ( x ) = q z  = 2 q   θ and a fluctuating (dynamic) part

(5.15)

96

5 WIND AND MOTION INDUCED LOADS

q y    q ( x , t ) =  q z  = Bq ⋅ v + Cae ⋅ r + K ae ⋅ r q   θ

(5.16)

where Bq ⋅ v is the dynamic loading associated with turbulence ( u and w ) in the oncoming flow, while Cae ⋅ r and K ae ⋅ r are motion induced loads associated with structural velocity and displacement. It is seen that linearity has been obtained, and thus, the theory is applicable in time domain as well as in frequency domain. The frequency domain amplitudes of the dynamic load are obtained by taking the Fourier transform throughout Eq. 5.16. Thus,

aq = Bq ⋅ av + ( iωCae + K ae ) ⋅ ar

(5.17)

T  a qθ     T  a rθ       

(5.18)

where:

aq ( x , ω ) = a q y 

a qz

ar ( x , ω ) = a ry 

a rz

av ( x , ω ) = [ a u

aw ]

T

and where i is the imaginary unit. Taking it for granted that the theory will primarily be applied in a modal frequency domain approach it is favourable to introduce two major improvements. First, for the purpose of frequency domain calculations it has been suggested to include frequency dependent flow induced dynamic loads, i.e. to replace Bq ( x ) in Eq. 5.12 with 2 ( D / B ) C D A yu  ρVB  Bq ( x , ω ) = 2C L A zu 2   2 B C M Aθ u 

( ( D / B ) C D′ − C L ) A yw  (C L′ + ( D / B ) C D ) A zw  ′ Aθ w B CM

(5.19)

 

where:

A m n (ω )

m = y , z ,θ  n = u , w

(5.20)

are the so-called cross sectional admittance functions. They are frequency dependent functions characteristic to the cross section in question. In general, they may

5.2 AERODYNAMIC DERIVATIVES

97

be determined from section model wind tunnel experiments, either directly from pressure tap measurements around the periphery of the cross section, or from time series of drag, lift and moment forces on the model that are otherwise used to determine mean load coefficients, in which case it is necessary to assume that the length scales of the fluctuating forces are identical to the appropriate length scales of the turbulent flow components. Cross sectional admittance functions have been theoretically developed for a thin airfoil by Sears [15], but since Sears solution is complex and contain cumbersome Bessel functions, approximate expressions, usually of the following type have been suggested (first by Liepmann [16])

A m n (ω ) =

m = y , z ,θ  n = u , w

1

(1 + a m n B ω / V )

bm n

(5.21)

where a m n and bm n are cross sectional dependent constants. As can be seen,

A m n (ω ) ≤ 1 A m n (ω = 0 ) = 1 lim A m n (ω ) = 0

ω →∞

     

(5.22)

and thus, its main effect is to filter off load contributions at high frequencies. (Other expressions may be expected for complex cross sections.) The second major improvement to the frequency domain application of the buffeting theory is to replace the content of Cae and K ae with the so-called aerodynamic derivatives. That is dealt with in the next chapter.

5.2

Aerodynamic derivatives

As derived from the buffeting theory Cae and K ae are given in Eqs. 5.13 and 5.14. They are three by three matrices containing all the eighteen coefficients that are required for a full frequency domain description of motion induced dynamic forces associated with structural velocity and displacement. The modal frequency domain counterparts to Cae and K ae are first fully presented in Eq. 4.62 in chapter 4.4. (Basic assumptions are given in Eq. 4.59. Mae is in the following considered negligible.) The essential theory presented below was first developed in the field of aeronautics and later made applicable to bridges by Scanlan & Tomko [17]. Following their notations, rather than the more general use of symbols shown in chapter 4.4, the frequency domain versions of Cae and

K ae are given by


98

Cae

 P1 =  H 5  A 5

P5 H1 A1

P2  H 2  A 2 

and

K ae

 P4 =  H 6  A 6

P3  H 3  A 3 

P6 H4 A4

(5.23)

The coefficients contained in Cae and K ae are then functions of the frequency of motion, the mean wind velocity and the type of cross section (and to some extent the initial or mean angle of incidence and the turbulence properties in the oncoming flow). Usually, they have been experimentally determined in wind tunnel aeroelastic section model tests, limited to vertical and torsion displacements. Since their main use lies in the detection of unstable motion at high wind velocities, the primary modal mass and stiffness properties of the section model will intentionally only contain the eigenfrequencies associated with the most onerous modes with respect to unstable structural oscillations. For a plate-like bridge cross section this is usually the lowest mode in torsion together with the shape-wise similar and lowest vertical mode. (Shape-wise similarity is required because the effect of aerodynamic coupling between the two modes is often important.) Since the along wind motion is absent in the section model, all terms associated with this direction must either be disregarded or taken from the quasi static buffeting theory (see Eqs. 5.25 and 5.26). The tests may be performed in three alternative ways. The original procedure was to extract the motion induced forces from the changes in resonance frequency and damping properties in transient (i.e. decay) recordings at various wind velocities under the conditions of pure vertical motion, pure torsion and finally combined vertical and torsion (see appendix C). Another procedure is to use a section model that undergoes forced oscillations at various wind velocities. From such a steady-state situation cross sectional forces are measured by pressure tap recordings on the surface of the model hull. Subtraction of the forces at zero motion will then render net motion induced effects. The third procedure is to perform ambient vibration tests, again at various wind velocities, and use the theory of system identification to extract the sought flow-structure interaction properties. It has been considered convenient to normalise Cae and K ae with ρ B 2ωi / 2 and ρ B 2ωi2 / 2 , where ωi is the in-wind (mean wind velocity dependent) resonance

frequency associated with the mode shape (number i ) from which they have been extracted. Thus,

Cae = where

ρB 2 2

ˆ ⋅ ωi (V ) ⋅ C ae

and

K ae =

ρB 2 2

2

⋅ ωi (V )  ⋅ Kˆ ae

(5.24)


ˆ C ae

 P1*  =  H 5*  *  B A 5

P5* H 1* B A1*

B P2*   B H 2*   B 2 A 2* 

and

Kˆ ae

 P4*  =  H 6*  *  B A 6

99 P6* H 4*

B A 4*

B P3*   B H 3*   B 2 A 3* 

(5.25)

It is the non–dimensional coefficients Pk* , H k* , A k* , k = 1 − 6 that are called the

aerodynamic derivatives. The values that emerge from the buffeting theory are obtained by comparison to Eqs. 5.13 and 5.14, rendering

 P1*  *  P2  *  P3  *  P4 P *  5 P *  6

H 1* H 2* H 3* H 4* H 5* H 6*

A1*   A 2*   A 3*   A 4*  A 5*   A 6* 

 D V  −2C D B B ω i (V )   0  2     C D′ D  V  = B  B ωi (V )   0   D   V  C L − C D′  B  B ωi (V  0 

D V  −  C L′ + C D  B  B ωi (V  0  V C L′   B ω (V i  0

)

−2C L

  ) 

V B ωi (V 0

2

)

)

    0  2  V    ′  CM  B ω (V )   i    0   V  −2C M B ωi (V )   0  ′ −C M

V B ωi (V

)

(5.26) As shown in Eq. 5.26, the aerodynamic derivatives will be functions of the reduced velocity V ωi (V ) B  . It should be noted that in the determination of the reduced velocity [or the non-dimensional resonance frequency ωî = B ωi (V ) / V ] the resonance frequency ωi (V ) is a function of the mean wind velocity, V . To start off with, i.e. at V = 0 , ωi (V = 0 ) is the eigen-frequency in still air conditions,. It is then only dependent

on the relevant structural properties. At V ≠ 0 the aerodynamic derivatives contained in K ae will have the effect of changing the total stiffness of the combined structure and flow system. This implies that the resonance frequency at V ≠ 0 is different from the initial value that was determined at V = 0 (or in vacuum). In general the consequence of this effect is that any response calculation involving the aerodynamic derivatives contained in K ae will demand iterations. However, under normal circumstances the effects of K ae will only be of significant importance in the velocity region at or immediately below an instability limit. At a characteristic mean wind velocity well below such an instability limit it is usually the aerodynamic derivatives contained in Cae that play the leading role, and the effects of the changes of ωi with increasing V to the determination of the aerodynamic derivatives are most often only of minor importance, especially as compared to other uncertainties in the theory (see further discussion in chapters 6.3 and 8). On the other hand, at or in the vicinity of an instability limit the flow


100

induced changes to the resonance frequency will in most cases be of great importance, and thus, for the determination of an instability limit this effect can usually not be ignored (see chapter 8). Aerodynamic derivatives for an ideal flat plate type of cross section were first developed by Theodorsen [28]. They are given by:

 H 1*  * H 2  * H 3  * H 4

 −2π FVî  A1*     π 1 + F + 4 GVˆ Vˆ i i A 2*   2 = * A 3   2π FVˆ − G 4 Vˆ i i   A 4*   π  1 + 4 GVî  2

(

)

(

(

) )

−

π

( π ( 2

8

π

   ˆ ˆ 1 − F − 4 GV i V i    FVî − G 4 Vî    π ˆ  GV i  2 −

2

FVî

)

)

(5.27)

where Vî = V  B ωi (V )  is the reduced velocity, and

 ωˆ  J ⋅ ( J 1 + Y 0 ) + Y1 ⋅ (Y1 − J 0 )  F i= 1   2  ( J 1 + Y 0 )2 + (Y1 − J 0 )2   J 1 ⋅ J 0 + Y1 ⋅ Y 0  ωî   G  = − 2 2  2   ( J 1 + Y 0 ) + (Y1 − J 0 ) 

(5.28)

are the real and imaginary parts of the so-called Theodorsen’s circulatory function. Their content J n (ωî 2 ) and Y n (ωî 2 ) , n = 0 or 1 , are first and second kinds of Bessel functions with order n , and ωî is the non-dimensional resonance frequency, i.e. ωî = B ωi (V ) / V = Vˆ −1 . The flat plate aerodynamic derivatives given in Eq. 5.27 are

plotted in Fig. 5.3.


Fig. 5.3

Flat plate aerodynamic derivatives (broken lines are the quasi- static values)

101


106

  ∆ x 2   2 ∆x  ˆ −  Co ∆ x = cos ⋅ exp )     qm (   3 λm D    3 λm D   

(5.34)

where m = z or θ , ωs = 2π fs , σˆ qm is the non-dimensional root mean square lift or torsion moment coefficient, bm is a non-dimensional load spectrum band width parameter, λm is a non-dimensional coherence length scale and ∆x is span–wise separation. In general, σˆ q z increases with increasing bluffness of the cross section, bz attains values between 0.1 and 0.3, while λz is typically in the order of 2 to 5. Similar properties may be expected of qθ . For the description of the characteristic motion induced load effects at “lock-in” Vickery & Basu [18, 19] have suggested that this may be accounted for by a negative motion dependent aerodynamic modal damping ratio, ζ aei , such that the total modal damping ratio associated with mode i is given by

ζ toti = ζ i − ζ aei

(5.35)

This is equivalent to the introduction of motion dependent aerodynamic derivatives as described in chapter 5.2 above. Adopting the notation given in Eqs. 5.24 and 5.25, it is the aerodynamic derivatives H 1* and A 2* that are responsible for aerodynamic damping exclusively effective in the across wind vertical (z) direction or in torsion (θ). Assuming that in the vicinity of a distinct vortex shedding type of response all other motion induced effects may be ignored, then

Cae ≈

0  ωi (V ) 0 2  0

ρB 2

0 H 1*

0

  0   B 2 A 2*  0

and

K ae ≈ 0

(5.36)

  σ 2  1 −  θ     aθ    

(5.37)

where

H 1*

= K az

  σ 2  1 −  z     az D    

and

A 2*

= K aθ

and where K a z and K aθ are the velocity dependent damping coefficients equivalent to those defined by Vickery & Basu [18, 19]. (However, if appropriate experimental evidence is available, there is no reason why Cae and K ae should not be full three by three matrices, also in the region of distinct vortex shedding excitation.) Assuming that ωi (V ) ≈ ωi (V = 0 ) , then the aerodynamic damping term in Eq. 5.35 may be taken

from Eq. 4.73, and thus,

5.3 VORTEX SHEDDING

ζ aei

  L exp   2m i ∫ φTi ⋅ φi d x i i  L    * 2 2 * 2 ∫ H 1 φz + B A 2 φθ d x  2  ρ B L exp = ⋅  2 2 2 4 m i ∫ φy + φz + φθ d x   L

C aei = = 2ω M

∫

φTi ⋅ Cae ⋅ φi d x

(

⇒ ζ aei

107

)

(

(5.38)

)

where m i =

M i

∫ φi

T

L

⋅ φi d x

=

M i

∫ ( φ y + φz 2

L

2

(5.39)

)

+ φθ2 d x

are the evenly distributed and modally equivalent masses associated with mode i . K am ( m = z or θ ) are the coefficients that account for the accelerating part of the motion induced load when V is close to V R i . Apart from being cross sectional characteristics, they are functions of V and the resonance frequency of the mode in question (see right hand side diagram in Fig. 5.5). a z D and aθ are quantities associated with the selflimiting nature of vortex shedding, i.e. they represent upper displacement or rotation limits at which the aerodynamic damping becomes insignificant. It should be noted that in Eq. 5.37 the damping coefficients are defined such that consistency is obtained with the general definition of aerodynamic derivatives in Eqs. 5.24 and 5.25 rather than the definition adopted by Vickery & Basu [18, 19]. [Thus, the K a z values given by Vickery & Basu in references [18, 19] are applicable in the expressions given above if they are multiplied by 4 ( D B ) . Vickery & Basu have not 2

given any recommendations regarding the K aθ coefficient.] It should also be noted that vortex shedding effects are to some extent dependent on the Reynolds number ( Re = VD / υ where υ = 1.5 ⋅ 10 −5 m2/s is the kinematic viscosity of air) and of the turbulence properties in the oncoming flow. Information about these effects is presented by Simui & Scanlan [4] and by Dyrbye & Hansen [5]. For a tubular cross section the Reynolds number effect is to change the point of flow separation, thus changing the Strouhal number as well as the load intensity. The main effect of turbulence is to broaden the band-width and disturb the size and coherence of the pressure fluctuations on the surface of the structure. Most structures are more prone to vortex induced oscillations in smooth flow.

108


Above, only the effects in the across wind direction and torsion have been included. In general, vortex shedding will also generate more or less narrow–banded load fluctuations in the along wind direction, but at a frequency twice that which occurs in the across wind direction and for most bridges at an insignificant load intensity.

110

6 WIND INDUCED STATIC AND DYNAMIC RESPONSE CALCULATIONS

design point of view the main focus is on quantifying the maximum value of the response that is most critical with respect to structural safety.

Fig. 6.2

Time and frequency domain representations

The flow is in general assumed Gaussian, stationary and homogeneous over a certain short term period T (e.g. 10 min), i.e. the response calculations are performed for a chosen design weather condition that is stable in time and space. If the mathematical transfer from flow properties to forces is linear and the structure is linear elastic, then the assumption of Gaussian and stationary properties also holds for any structural response quantity. Thus, any response quantity (e.g. a displacement) may be described by its mean

6.1 INTRODUCTION

111

value and probability density distribution, as shown to the right in Fig. 6.2.a. Its maximum value at position x r is then given by

rm a x ( x r ) = r ( x r ) + k p ⋅ σ r ( x r )

(6.1)

where r ( x r ) is the mean value, k p is the peak factor that depends on the type of process (see chapter 2.4) and σ r ( x r ) is the standard deviation of the fluctuating part of the response. The mean value r ( x r ) may be obtained from simple static equilibrium conditions. The standard deviation of the fluctuating part of the response σ r ( x r ) may either be obtained from a time domain integration of the dynamic load effects from the fluctuating flow field and possible vortex shedding, or from a modal approach in frequency domain. The former alternative is computationally a demanding task, as it requires the time domain simulation of a wind field that is usually broad banded and spatially un-correlated (such a simulation procedure is shown in appendix A). In the following it is the alternative of a modal frequency domain approach that is presented. As illustrated in Fig. 6.2.b, the main steps involve the transfer from a wind field crossspectral density via a corresponding modal load spectrum to the final sought response spectrum. The area under the response spectrum is then the variance σ r2 of the response. As shown in Eq. 1.6 (and illustrated in Fig. 1.3.b) the cross sectional displacement at a position x r that has been chosen for the relevant response calculation is in general a vector containing three components: ry in the along wind horizontal direction, rz in the across wind (for a bridge) vertical direction and the cross sectional rotation rθ . Since these describe a combined cross sectional displacement in a plane perpendicular to the span, the peak factor in Eq. 6.1 is equally applicable to each of the components, and thus

ry  ry      rm a x ( x r ) =  rz  =  rz  + k p r     θ  m a x rθ 

σ ry    ⋅ σ rz    σ rθ 

(6.2)

As mentioned above (and further discussed in chapter 1), the wind induced response of a slender structure is assumed stationary, and then the total response may be split into a mean (static) and a fluctuating (dynamic) part. What can in general be expected in the case of a slender structure is illustrated in Fig. 6.3. The static part is proportional to the mean velocity pressure, i.e. to the mean wind velocity squared, until motion induced forces may reduce the total stiffness of the combined structure and flow system, after which the static response may approach an instability limit (torsion divergence). The dynamic part of the response may conveniently be separated into three mean wind velocity regions. Vortex shedding effects will usually occur at fairly low mean wind velocities, buffeting will usually be the dominant effect in an intermediate velocity

6.2 THE MEAN VALUE OF THE RESPONSE

113

aerodynamic derivatives and their contribution to total stiffness and damping it is assumed that the effect of changing resonance frequencies may be ignored. For the response calculations in this chapter motion induced load effects may then be taken at a reduced velocity V /(ωi B ) where ωi is the predetermined resonance frequency based on structural properties alone and at V = 0 . Otherwise, iterations are required. Thus, it is assumed that the response calculations are not taken in close vicinity to a motion induced instability limit. However, in the determination of an instability limit as shown in chapter 8, this effect can not be ignored, and ωi will be taken at the relevant critical wind velocity, V cr . Thus, the determination of V cr in chapter 8 will demand iterations.

6.2

The mean value of the response

The mean value of the response is the load effects of the mean flow induced load as defined in Eq. 5.11. It may readily be calculated according to standard static equilibrium type of procedures in structural mechanics. Such procedures are in general mathematically formulated within a finite element type of description where the solution strategy is based on the displacement method, i.e. for a chosen discrete model containing N number of nodes the mean displacement vector r is obtained from K⋅r =R

(6.3)

where K is the static stiffness matrix and R is the mean load vector. A line like structure will in general be modelled by beam or beam-column type of elements, in which case there will usually be six degrees of freedom in each node (as illustrated in Fig. 6.4.a). Thus, r and R are 6 ⋅ N by one vectors and K is a 6 ⋅ N by 6 ⋅ N matrix. Herein, the establishment of K and the ensuing strategy for the calculation of r will not be further pursued. However, the establishment of R is presented below. Let us consider a typical finite element type of modelling with six load components in each node. According to Eq. 5.11 the mean value of the evenly distributed load on an element is given by

q y    ρV 2 q ( x ) = q z  = ⋅ bq 2 q   θ

where

 DC D  bq =  B C L  2 B CM

    

(6.4)

6.2 THE MEAN VALUE OF THE RESPONSE

115

At an arbitrary node p the load contribution from an adjoining Fig. 6.4.b and c) is then

Q pm

element m (see

Q y     ρV 2  L m L = Q z  = qm ( x ) ⋅ m =  ⋅ ⋅ b qm  2  2 Q   p 2  θ 

(6.5)

where L m is the element length, bqm is the bq vector that contains the properties associated with element m , and where it has for simplicity been assumed that the nodal discretisation is such that q may be taken constant within the length of the element (otherwise, L m 2 may be replaced by the result of a simple span-wise integration). Comparing the situation in Fig. 6.4.b and c to the general definition of external load components in Fig. 6.4.a, it is then seen that the contribution from Q pm to the load vector is R pm = [ R1

R2

R3

R4

R5

R 6 ]p T

m

= 0 Q y

−Qθ

Qz

T

0 0  pm

(6.6)

if m is horizontal, and R pm = [ R1

R2

R3

R4

R5

R 6 ]p T

m

=  −Q z

0 0 0 −Qθ 

Qy

T pm

(6.7)

if m is vertical. Thus,  ρV 2 R pm = θm ⋅ Q pm =   2 

 Lm ⋅ θ m ⋅ b qm  ⋅ p 2

(6.8)

where

θm

0 1  0 = 0 0  0

0 0 1 0 0 0

0 0  0  if m is horizontal, and −1  0  0 

θm

0 1  0 = 0 0  0

−1 0 0 0 0 0

0 0  0  0 0  −1 

if m is vertical.

The six by one load vector R p in node p is then given by the sum of the contributions from all adjoining elements, i.e.

116


R p = [ R1

R2

R3

R4

R5

R 6 ] p = ∑ R pm T

(6.9)

m

T

and the total 6 ⋅ N by one load vector is given by: R = R1 " R p " R N  .

6.3

Buffeting response

As previously discussed in chapter 4, for practical reasons it is in the following distinguished between three cases. First a case of single mode single component response will be shown. This will render a suitable solution if eigen-frequencies are well separated and there is insignificant structural or flow induced coupling between horizontal, vertical and torsion displacement components. Second, a case of single mode three component response will be shown. This is a suitable solution strategy if there is significant structural or flow induced coupling between any of the three displacement components, and if eigen-frequencies are still well separated. Finally, a full multi mode approach is presented. The buffeting load is given in chapter 5.1. As shown in Eq. 5.8 (see also Eqs. 5.15 and 5.16), it comprises a time invariant mean part q ( x ) , previously dealt with in chapter 6.2 above, and a fluctuating part

q ( x , t ) = Bq ⋅ v + Cae ⋅ r + K ae ⋅ r

(6.10)

that contains a flow induced contribution Bq ⋅ v and two motion induced parts Cae ⋅ r and K ae ⋅ r . The content of Eq. 6.10 is defined in Eqs. 5.9 – 5.14. It is applicable in time domain as well as in frequency domain. Improved frequency domain counterparts to Bq , Cae and K ae are given in Eqs. 5.19, 5.24 and 5.25. As shown in chapter 4.2 – 4.4, in a

modal frequency domain solution the flow induced part of the load (i.e. the modal versions of Cae ⋅ r and K ae ⋅ r ) are moved to the left hand side of the equilibrium equation and included in the modal frequency-response-function. Thus, the development of a modal buffeting load needs only consideration of the flow induced part

6.3 BUFFETING RESPONSE

117

Bq ⋅ v , while the motion induced parts need consideration in the development of the

modal frequency-response-function.

Single mode single component buffeting response calculations The response spectrum of an arbitrary displacement component at span–wise position x r due to excitation in a corresponding mode shape number i is given in Eqs. 4.28 – 4.30. The variance of the displacement response at x r is then obtained by frequency domain integration, i.e. ∞

σ r2i ( x r ) = ∫ S ri ( x r , ω ) d ω =

φi2 ( x r )

0

where S Q (ω ) = lim and a Q

i

1

T →∞ π T

i

K i2

(a

*

Q i

∞

2

⋅ ∫ Hˆ i (ω ) ⋅ S Q (ω ) d ω i

(6.11)

0

⋅ aQ

i

)

(6.12)

is the Fourier amplitude of the appropriate flow induced modal loading

component q y , q z or qθ . The modal stiffness K i and the modal frequency-responsefunction Hˆ i (ω ) are defined in Eqs. 4.19 and 4.24. As shown in Eq. 4.24, any motion induced load effects are included in Hˆ i (ω ) . Let us for simplicity consider the displacement response in the along wind horizontal direction ry at x r , and develop its variance contribution from one of the predominantly T

y–modes, φi ≈ φ y 0 0  , with corresponding eigen-frequency ω i = ω y (e.g. the contribution from the y-mode with lowest eigen-frequency). The flow induced modal load is then given by (see Eqs. 4.19 and 5.12) Q y (t ) =

∫

φy ( x ) ⋅ q y ( x ,t ) d x

∫

φ y ( x ) ⋅ 2

L exp

=

ρVB 2

⋅

L exp

 D D C D ⋅ u ( x , t ) +  C D′ − C L B  B

   ⋅ w ( x ,t ) d x  

(6.13)

where L exp is the flow exposed part of the structure. Taking the Fourier transform on either side renders


118 a Q =

y

(ω ) = ∫

L exp

ρVB 2

∫

⋅

φy ( x ) ⋅ a q y ( x ,ω ) d x  D D C D ⋅ a u ( x , ω ) +  C D′ − C L B B 

φ y ( x ) ⋅ 2

L exp

   ⋅ aw ( x , ω ) d x  

(6.14)

and thus, the modal load spectrum is given by

S Q

y

2  1   ρVB  lim  ∫ φy   2  T →∞ π T  L exp 

(ω ) = 

  ⋅  ∫ φy  L exp

 D D * 2 B C D a u +  B C D′ − C L  

 D D 2 B C D a u +  B C D′ − C L  

  *  a d x  w     

    a d x  w     

(6.15)

Acknowledging that 1  * a m ( x1 , ω ) ⋅ a n ( x 2 , ω )  T →∞ π T 

S m n ( ∆x , ω ) = lim

where

m  = u ,w n

(6.16)

and assuming that the cross spectra between flow components are negligible, i.e. that

S u w ( ∆x , ω ) = S w u ( ∆x , ω ) ≈ 0

(6.17)

then S Q

y

 ρV 2 B  ⋅ J y (ω )  (ω ) =   2 

2

(6.18)

where

J y2 (ω ) =

∫∫

2  D  S ( ∆x , ω ) CD I u  uu 2 σu   B

φ y ( x1 ) ⋅ φ y ( x 2 ) ⋅  2

L exp

 D +  C D′ − C L  B

2   S w w ( ∆x , ω )  I  d x1 d x 2  w σ w2   

(6.19)

is the joint acceptance function containing the span-wise statistical averaging of variance contributions from the fluctuating u and w flow components. I u and I w are the corresponding turbulence intensities and ∆x = x1 − x 2

is the spatial (span-wise)

separation. Combining Eqs. 6.11 and 6.18, using K y = ω y2 M y , and introducing the modally equivalent and evenly distributed mass


m y = M y

119

∫ φy d x = ∫ m yφy d x ∫ φy d x 2

2

L

L

2

(6.20)

L

then the following expression is obtained for the standard deviation of the dynamic response in the along wind y direction

σ y ( x r ) = φy ( x r )

ρB 3  V ⋅ ⋅ 2m y  B ω y

1/2

2

  ∞ 2 2  ⋅  ∫ Hˆ y (ω ) ⋅ Jˆ y (ω ) d ω     0 

Jˆ y (ω ) = J y / ∫ φ y2 d x

where:

(6.21)

(6.22)

L

The non-dimensional frequency response function is given in Eq. 4.25. Neglecting any aerodynamic mass effects and introducing the notation given in Eqs. 5.24 and 5.25, it is then given by 2   ω  ω ˆ  H y (ω ) = 1 − κ ae y −   + 2i ζ y − ζ ae y ⋅  ωy   ωy     

(

)

−1

(6.23)

where:

 ρB 2 2 *  ω y P4 ∫ φ y2 d x    2  L exp  K ae    y 2 2    φ y2 d x ω y m y ∫ φ y d x ∫  2 κ 2  ae y   ω y M y    ρ B L exp L  = =  = m ⋅ 2 ζ ae y   C ae   ρ B φ y2 d x y * 2 ∫     y ω P φ d x   y 1 ∫ y L   2ω y M y   2 L exp   2ω m φ 2 d x  y y∫ y   L

1 *   2 P4  ⋅  1 P *   4 1 

(6.24)

Similarly, the standard deviation of the dynamic response in the z direction and in torsion are given by 2

1/2

σ z ( x r ) = φz ( x r )

∞  2 ρB 3  V   ˆ 2 ⋅ ⋅  ⋅  ∫ H z (ω ) ⋅ Jˆ z (ω ) d ω  2m z  B ωz   0 

σ θ ( x r ) = φθ ( x r )

∞  2 ρB 4  V   ˆ 2 ⋅ ⋅  ⋅  ∫ H θ (ω ) ⋅ Jˆθ (ω ) d ω  2m θ  B ωθ   0 

2

(6.25) 1/2

(6.26)

120


where:

          

∫ m z φz d x 2

m z = M z / ∫

φz2 d x

=

L

∫ φz d x 2

L

L

∫ m θ φθ d x 2

m θ = M θ / ∫ φθ2 d x = L

L

∫ φθ d x 2

L

(6.27)

and where the joint acceptance functions are given by   Jˆ z (ω ) =  ∫∫ φz ( x1 ) ⋅ φz ( x 2 ) ⋅  2C L I u    L exp

(

S u u ( ∆x , ω )

)

2

σ u2

(6.28)

12

  D   S ( ∆x , ω )  +  C L′ + C D  I w  w w 2  d x1 d x 2  B  σw      2

  Jˆθ (ω ) =  ∫∫ φθ ( x1 ) ⋅ φθ ( x 2 ) ⋅  2C M I u    L exp

(

)

2

∫ φz d x 2

L

S u u ( ∆x , ω )

σ u2

(6.29)

12

′ Iw ) + (C M

2

 S w w ( ∆x , ω )   d x1 d x 2  2 σw  

∫ φθ d x 2

L

The corresponding frequency response functions (see Eqs. 4.24, 4.68 and 4.69) are given by 2  ω  ω Hˆ z (ω ) = 1 − κ aez −   + 2i ζ z − ζ aez ⋅  ωz    ωz   

(

)

−1

2  ω  ω  Hˆ θ (ω ) = 1 − κ aeθ −   + 2i ζ θ − ζ aeθ ⋅ ωθ    ωθ   

(

where:

)

−1

        

(6.30)


κ aez  ζ aez

∫

φz2 d x  1

∫

φθ2 d x  1

*  ρ B L exp 2 H 4  ⋅ ⋅ =  2 m z  ∫ φz d x  1 H 1*  L 4  2

 A 3* 

κ aeθ  ρ B 4 L exp 2 ⋅ ⋅  =  2 m θ ζ aeθ  ∫ φθ d x  1 A 2*  L 4 

121           

(6.31)

Example 6.1: The volume integral in the joint acceptance functions above, e.g. as first defined in Eq. 6.19 or as normalised versions given in Eqs. 6.22, 6.28 and 6.29, may in general be expressed by J r2m rn =

where:

L exp L exp

g rm rn ( x1 , x 2 ) d x1d x 2

∫ ∫ 0

0

g rm rn ( x1 , x 2 ) = G rm ( x1 ) ⋅ G rn ( x 2 ) ⋅ψ k k ( ∆x ) ,

m  = y , z ,θ n  k = u , w . It will in most cases

demand a fine mesh, particularly in the region of small separation ∆x = x1 − x 2 . The reason for this is that ψ k k , is usually rather steep close to zero, and thus, g rm rn ( x1 , x 2 ) will rapidly drop in the region close to a diagonal plane through x1 = x 2 . This difficulty may readily be overcome by adopting Dyrbye & Hansen’s [21] following procedure for turning a volume integral back into two line integrals. The position coordinates x1 and x 2 are interchangeable, and therefore g rm rn ( x1 , x 2 ) will be symmetric about the plane through x1 = x 2 . Thus, J

2 rm rn

L exp

=2

∫ 0

 L exp   g rm rn ( x1 , x1 − ∆x ) d x1  d ∆ x ∫  ∆x   

Introducing the notation x1 = x + ∆x and g rm rn ( x1 , x1 − ∆ x ) = Grm ( x1 ) ⋅ Grn ( x1 − ∆ x ) ⋅ψ k k ( ∆ x ) = Grm ( x + ∆ x ) ⋅ Grn ( x ) ⋅ψ k k ( ∆ x ) then the following is obtained: J r2m rn = 2

L exp

∫ 0

 L exp − ∆x   G rm ( x + ∆ x ) ⋅ G rn ( x ) d x  ⋅ψ k k ( ∆ x ) d ∆ x ∫  0   

It is usually convenient to introduce the normalised coordinate xˆ = x / L exp and separation

∆xˆ = ∆ x / L exp . Thus, in a normalised format the joint acceptance function is given by 1 1 − ∆xˆ  J r2m rn = 2 L2exp ∫  ∫ G rm ( xˆ + ∆xˆ ) ⋅ G rn ( xˆ ) d xˆ  ⋅ψ k k ( ∆ xˆ ) d ∆ xˆ  0   0

Let for instance G rm ( x1 ) = x1 / L exp and G rn ( x 2 ) = x 2 / L exp , then


122

1 1 − ∆xˆ 1  1 xˆ + ∆ xˆ xˆ 3 ⋅ J r2m rn = 2 L2exp ∫  ∫ d xˆ  ⋅ψ k k ( ∆ xˆ ) d ∆ xˆ = ∫ 2 − 3 ( ∆ xˆ ) + ( ∆ xˆ ) ψ k k ( ∆ xˆ ) d ∆ xˆ   L L 3  0  exp exp 0 0 The solutions to a good number of cases have been shown by Dyrbye & Hansen [21] and by Davenport [14], who has also developed simple approximate expressions. The most common cases are graphically illustrated in appendix B.

Example 6.2: Let us consider a typical single mode single component situation, where the three modes k , m , n φ k = φ y

0 0 

T

φm = [0 φz

0]

φn = [0 0 φθ ]

T

T

with corresponding eigen-frequencies ω y , ωz , ωθ have been singled out for a response calculation. Since the main girder cross section of many bridges are close to a flat plate, the load coefficient properties C D′  CD    D C L′  ≠ 0 , and CL  ≈ 0 C D C L′ B   ′ CM CM   are frequently encountered in bridge engineering. In that case  D  J y2 (ω ) =  2 C D I u   B 

J z2 (ω ) = (C L′ I w )

2

2

∫∫

φ y ( x1 ) ⋅ φ y ( x 2 ) ⋅

2

σu

L exp

S w w ( ∆x , ω ) d x1d x 2 2

∫∫

φz ( x1 ) ⋅ φz ( x 2 ) ⋅

∫∫

φθ ( x1 ) ⋅ φθ ( x 2 ) ⋅

L exp

′ Iw ) J θ2 (ω ) = (C M

S u u ( ∆x , ω ) d x1 d x 2 2

L exp

σw

S w w ( ∆x , ω ) d x1d x 2 2

σw

Introducing: ˆ S u u ( ∆x , ω ) = S u (ω ) ⋅ Co u u ( ∆x , ω ) ,

ˆ S w w ( ∆x , ω ) = S w (ω ) ⋅ Co w w ( ∆x , ω ) ,

K n = ωn2 ⋅ M n = ωn2 ⋅ m n ∫ φn2 d x , n = y , z or θ L

and the non–dimensional joint acceptance functions 12

  ˆ  Jˆ y (ω ) =  ∫∫ φ y ( x1 ) ⋅ φ y ( x 2 ) ⋅ Co u u ( ∆ x , ω )d x 1 d x 2   L exp  

∫ φy d x 2

L 12

  ˆ  Jˆ z (ω ) =  ∫∫ φz ( x1 ) ⋅ φz ( x 2 ) ⋅ Co w w ( ∆ x , ω )d x 1 d x 2   L  exp 

∫ φz d x 2

L


123 12

  ˆ  , Jˆθ (ω ) =  ∫∫ φθ ( x1 ) ⋅ φθ ( x 2 ) ⋅ Co ∆ x ω d x d x ( ) ww 1 2    L exp 

∫ φθ d x 2

L

then the ry , rz and rθ response spectra are given by (see Eq. 4.30, 6.18, 6.19, 6.28 and 6.29) 2

S ry

 ρ B 2D (ω , x r ) = φy ( x r ) ⋅ my 

S rz

2   S (ω ) ρB 3  V  w (ω , x r ) = φz ( x r ) ⋅   ⋅ C L′ I w ⋅ Hˆ z (ω ) ⋅ Jˆ z (ω )  ⋅ 2 m z  B ωz    σ w2  

 V ⋅  B ωy 

2  S ω  ( )  ⋅ C D I u ⋅ Hˆ y (ω ) ⋅ Jˆ y (ω )  ⋅ u 2   σu   2

2

2   S (ω ) ρB 4  V  ′ I w ⋅ Hˆ θ (ω ) ⋅ Jˆθ (ω )  ⋅ w 2 S rθ (ω , x r ) = φθ ( x r ) ⋅  ⋅CM 2m θ  B ωθ    σw   Integrating across the entire frequency domain, the following response standard deviations are obtained:

σ ry ( x r ) = φ y ( x r ) ⋅ σ rz ( x r ) = φz ( x r ) ⋅

ρ B 2D m y

 V ⋅ CD I u ⋅   B ωy 

12

2

  ∞ 2 S (ω )  ⋅  ∫ Hˆ y (ω ) ⋅ u 2 ⋅ Jˆ y2 (ω ) d ω    σu   0

12

 2 S (ω )  V  ∞ ˆ w ⋅ C L′ I w ⋅  ⋅ Jˆ z2 (ω ) d ω   ⋅  ∫ H z (ω ) ⋅ 2 2m z σw   B ωz   0 2

ρB 3

12

 2 S (ω )  V  ∞ ˆ w ′ Iw ⋅ ⋅CM ⋅ Jˆθ2 (ω ) d ω   ⋅  ∫ H θ (ω ) ⋅ 2 2m θ σw   B ωθ   0 Let us focus exclusively on the response in the y (drag) direction, and consider a simply supported horizontal beam type of bridge with span L = 500m that is elevated at a position z f = 50 m . Let

σ rθ ( x r ) = φθ ( x r ) ⋅

2

ρB 4

us for simplicity assume that the relevant mode shape φ y ( x ) = sin (π x L ) and that x r = L 2 , in which case φ y ( x r ) = 1 . Let us also assume that the entire span is flow exposed, i.e. L exp = L , and adopt the following wind field properties: I u = σ u V = 0.15 1) the turbulence intensity 2) the integral length scale: 3) the auto spectral density:

4) the normalised co-spectrum:

xf

(

L u = 100 ⋅ z f 10

S u (ω )

σ u2

=

(see Eq. 3.14)

)

0.3

1.08 ⋅

xf

(1 + 1.62 ⋅ ω ⋅

= 162m Lu V xf

Lu V

(see Eq. 3.36),

)

53

ˆ Co u u (ω , ∆x ) = exp ( −C u x ⋅ ω ⋅ ∆x V

(see Eq. 3.25)

)

(see Eq. 3.41)

where Cu x = Cu y f = 9 / ( 2π ) ≈ 1.4 . Let us allot the following values to the remaining constants that are necessary for a numerical calculation of σ ry ( x r = L 2 ) :


124

ρ (kg/m3)

CD

B (m)

D (m)

m y (kg/m)

ω y (rad/s)

ζy

1.25

0.7

20

4

10000

0.4

0.005

Since m y is constant along the span, then the modally equivalent and evenly distributed mass m y = m y . Finally, let us adopt quasi–static values to the aerodynamic derivatives, in which case

κ ae y = 0 and the aerodynamic damping ζ ae y is given by (see Eqs. 5.26 and 6.24) ζ ae y =

ρB 2 4 m y

P1* =

ρB 2 

D V  −2 C D 4 m y  B B ωy

 ρ DC DV ≈ −4.375 ⋅ 10 −4 ⋅ V =−  2 m ω y y 

The non-dimensional joint acceptance function Jˆ y may readily be obtained by numerical calculations. However, as shown by Davenport [14], in many cases closed form solutions may be ˆ obtained. The situation that φ y ( x ) = sin (π x L ) and Co u u (ω , ∆x ) is a simple exponential function is such a case. Substituting x1 = x , x 2 = x + ∆x , xˆ = π x L , ∆xˆ = ∆x L

and

ωˆ = Cu x ω L V , then Jˆ y2 =

ˆ φ y ( x1 ) ⋅ φ y ( x 2 ) ⋅ Co u u ( ∆x , ω ) d x1 d x 2

∫∫

L exp L exp

∫

=2

0

L exp

=2

∫ 0

 2   ∫ φy d x    L 

2

 L exp −∆x  ˆ   Co + ∆ ⋅ φ x x φ x d x ( ) ( ) ( ∆x , ω ) d ∆x y y  ∫0  uu  

L 2   ∫ φy d x    0 

 L exp −∆x  π π  sin ( x + ∆x ) ⋅ sin xd x  ⋅ exp ( − Cu x ω∆x V ) d ∆x ∫  0  L L  

2

L  π  ∫ sin 2 xd x    L 0 

2

Using that sin 2α = 2 sin α cos α and that sin (α + β ) = sin α ⋅ cos β + cos α ⋅ sin β , then this may be expanded into  L exp −∆x π π 1 2π    C ω∆x   2 π ∫  ∫  cos L ∆x ⋅ sin L x + 2 sin L ∆x ⋅ sin L x  d x  ⋅ exp  − u xV  d ∆x 0  0  π (1 −∆xˆ ) π (1 −∆xˆ ) 1  π π 8 1 ˆ xˆ + sin ∆xˆ ∫ sin 2 xd ˆ xˆ  ⋅ exp ( −ωˆ ∆xˆ ) d ∆xˆ = ∫  cos ∆xˆ ∫ sin 2 xd   L L π0 2 0 0  

8 Jˆ y2 = 2 L

L exp

1

8 π 1 Jˆ y2 = ∫  (1 − ∆xˆ ) cos π∆xˆ − cos π∆xˆ ⋅ sin 2π (1 − ∆xˆ ) − sin π∆xˆ ⋅ cos 2π (1 − ∆xˆ )  4 π 0 2 +

1  sin π∆xˆ  exp ( −ωˆ ∆xˆ ) d ∆xˆ 4 

Using that sin α ⋅ cos β − cos α ⋅ sin β = sin (α − β ) and sin ( −α + 2π ) = − sin α this simplifies into


125

1 1 1   Jˆ y2 = 4 ∫ (1 − ∆xˆ ) cos π∆xˆ + sin π∆xˆ  exp ( −ωˆ ∆xˆ ) d ∆xˆ = 4  ∫ cos π∆xˆ ⋅ exp ( −ωˆ ∆xˆ ) d ∆xˆ π   0 0 1

− ∫ ∆xˆ ⋅ cos π∆xˆ ⋅ exp ( −ωˆ ∆xˆ ) d ∆xˆ + 0

1

π



1

∫ sin π∆xˆ ⋅ exp ( −ωˆ ∆xˆ ) d ∆xˆ  

0

  (1 − ∆xˆ ) exp ( −ωˆ ∆xˆ )   = 4  ⋅ ( −ωˆ cos π∆xˆ + π sin π∆xˆ )  2 2 ωˆ + π 0   1

  exp ( −ωˆ ∆xˆ ) ˆ 2 ω − π 2 cos π∆xˆ − 2π sin π∆xˆ + 2 2 2 ˆ ω π + 

(

−

)

((

)

1

)

    0

1   1  exp ( −ωˆ ∆xˆ ) ˆ ˆ ˆ sin x cos x ⋅ ∆ + ∆ ω π π π ( )    π  ωˆ 2 + π 2  0 

Thus, the following is obtained: ⇒ Jˆ y2 (ωˆ ) = 4 ⋅ψ (ωˆ )

  ˆ )  ωˆ 2 1 + exp ( −ω ψ (ωˆ ) =  2 + 2π 2 2  ωˆ 2 + π 2   ωˆ + π 

where

(

)

The standard deviation of the dynamic response at x r = L 2 is then given by 12

σ ry ( L 2 ) = 3.28 ⋅10

−4

∞  2 S (ω ) ⋅ V  ∫ Hˆ y (ω ) ⋅ u 2 ⋅ Jˆ y2 (ωˆ ) d ω  σu  0  2

where S u (ω ) σ u2 , Jˆ y2 (ωˆ ) and ωˆ are defined above, and where

(

)

−1

2 Hˆ y (ω ) = 1 − (ω 0.4 ) + i 0.005 + 4.375 ⋅ 10 −4 ⋅ V ⋅ ω 0.4    The chosen single point spectral density and corresponding normalised co–spectrum of the turbulent u component are shown on the top left and right hand side diagrams in Fig. 6.5. The non-dimensional frequency response function and the squared normalised joint acceptance functions are shown on the lower left and right hand side diagrams in Fig. 6.5. The response spectrum of the along wind ry component at x r = L 2 and V = 40 m / s is shown in Fig. 6.6.

As can be seen, it contains a broad banded background part and a narrow banded resonant part at ω = 0.4 rad / s . The standard deviation of the dynamic response at x r = L 2 is plotted versus the mean wind velocity in Fig. 6.7. [It should be noted that the effect of aerodynamic damping is considerable (see Example 6.3), and that the validity of the quasi-static theory may be limited.]


126

Fig. 6.5

Fig. 6.6

Top left and right: single point u spectrum and corresponding normalised co-spectrum, lower left and right: frequency response function and joint acceptance function

Response spectrum of ry displacements at x r = L 2 and V = 40 m / s


Fig. 6.7

127

The standard deviation of the dynamic response at x r = L 2 versus the mean wind velocity

Single mode three component buffeting response calculations The solution to this case is given in chapter 4.3, see Eqs. 4.47 and 4.48. What remains from the development in chapter 4.3 is to expand on the modal load spectrum S Q using i

the results from chapter 5.1. As shown above (see Eq. 6.4 and ensuing discussion), the flow induced buffeting part of the fluctuating load is

q y ( x , t )    ˆ ⋅v  q z ( x , t )  = Bq ( x ) ⋅ v ( x , t ) = ( ρVB / 2 ) ⋅ B q q ( x , t )  θ  

ˆ and v are defined in Eqs. 5.9 and 5.12. Thus (see Eq. 4.39) where B q

(6.32)


128

ρVB Q i (t ) = 2 where φi ( x ) = φ y

∫

(

)

ˆ ⋅ v dx φTi ⋅ B q

L exp

(6.33)

T

φθ  . The Fourier transform of Eq. 6.33

φz

i

ρVB

a Q (ω ) =

2

i

∫

(

)

ˆ ⋅ a dx φTi ⋅ B q v

L exp

(6.34)

where

av ( x , ω ) = [ a u

aw ]

T

(6.35)

contains the Fourier amplitudes of the u and w components. This will then render the following modal load spectrum S Q (ω ) = lim

1

T →∞ π T

i

(a

* Q i

⋅ a Q

i

)

T      1  T ˆ ⋅ a* d x  ⋅  ˆ ⋅ a d x   = φT ⋅ B  ∫ φi ⋅ B q v q v  lim  L ∫ i    2  T →∞ π T   L exp exp      

 ρVB 

2

(

2

)

(

)

 ρVB  ˆ ( x ) ⋅ S ( ∆x , ω ) ⋅ B ˆ T (x ) ⋅ φ (x )d x d x = ⋅ ∫∫ φTi ( x1 ) ⋅ B 1 2 2 1 2 q v q i   2  L exp

{

}

(6.36) where 1  * 1 av ( x1 , ω ) ⋅ aTv ( x 2 , ω )  = lim  T →∞ π T T →∞ π T

Sv ( ∆x , ω ) = lim

 a u* a u  * a w a u

a u* a w   S u u S u w  =  a w* a w  S w u S w w  (6.37)

This is greatly simplified if the cross spectra between flow components are negligible, i.e. S u w = S w u ≈ 0 , see Eq. 6.17. Then  ρV 2 B  ⋅ J i (ω )  S Q (ω ) =  i  2 

2

(6.38)

where: J i2 =

∫∫

L exp

{

}

ˆ ∆x , ω )  ⋅ B ˆ ( x ) ⋅ I2 ⋅ S ˆT x φTi ( x1 ) ⋅ B ⋅ φi ( x 2 ) d x 1 d x 2 q 1 v v(  q ( 2)

(6.39)


129

is the joint acceptance function, and where

Iv = d iag [ I u

Iw ]

ˆ ( ∆x , ω ) = d iag S / σ 2 S v  uu u

S w w / σ w2 

    

(6.40)

Introducing the modal stiffness K i = ωi2 M i and defining

(

)

m i = M i / ∫ φTi ⋅ φi d x L

(6.41)

then from Eqs. 4.47 and 4.48 the following standard deviations of displacement responses at x r are obtained φ y ( x r )  σ y    ρB 3   σ φ x = ( )  z r   z   2m i σ   θ i φθ ( x r ) i

1/2

2  2  V  ∞ ˆ 2 ⋅  ⋅  ∫ H i (ω ) ⋅ Jˆ i (ω ) d ω   B ωi   0 

Jˆ i =

where:

Ji

∫

φTi

(6.42)

(6.43)

⋅ φi d x

L

Again, neglecting any aerodynamic mass and introducing the notation given in Eqs. 4.25, 4.40 and 5.25, then the frequency response function is given by 2  ω ω ˆ  H i (ω ) = 1 − κ aei −   + 2i ζ i − ζ aei ⋅  ωi    ωi    φT Kˆ φ d x

(

where:

κ aei =

ρB 2 2m i

∫

⋅

L exp

ζ aei =

ρB

4 m i

T i

∫

⋅

i

ae

i

(φ

T i

L exp

L

(6.45)

i

)

ˆ φ dx C ae i

∫ ( φi φi ) d x T

(6.44)

)

∫ (φ φ ) d x

L

2

(

)

−1

(6.46)

130


As explained in chapter 4.3 (see Eq. 4.41), only diagonal Kˆ ae and Cˆ ae will maintain the presupposition that no modal coupling will occur. Flow induced coupling will occur if ˆ are not diagonal. Kˆ ae and C ae

Multi mode buffeting response calculations The general solution to a multi mode approach is given by the three by three response matrix shown in Eqs. 4.80 – 4.82. The corresponding three by three response covariance matrix

 σ2  ry ry Covrr ( x r ) = Cov rz ry  Cov rθ ry

Cov ry rz

σ r2z rz Cov rθ rz

Cov ry rθ   Cov rz rθ   σ r2θ rθ  

(6.47)

which contains the variance of each response displacement component ry , rz and rθ at

x = x r on its diagonal and cross covariance on its off-diagonal terms, is obtained by frequency domain integration. Thus, ∞ ∞ *  ˆ (ω ) S (ω ) H ˆ T (ω ) d ω  ΦT ( x ) (6.48) Covrr ( x r ) = ∫ Srr ( x r , ω ) d ω = Φr ( x r )  ∫ H η η r r Qˆ 0  0 

where Hˆ η (ω ) and SQˆ (ω ) are N m od by N m od matrices given in Eqs. 4.69 and 4.75, and Φr ( x r ) is a three by N m od matrix defined in Eq. 4.79. What remains is to bring the

results from chapter 5.1 into Hˆ η (ω ) and SQˆ (ω ) . Disregarding any aerodynamic mass effects, the frequency response matrix Hˆ η (ω ) in Eq. 4.69 is reduced to 2     1  1    ˆ Hη (ω ) = I − κ ae −  ω ⋅ d iag    + 2iω ⋅ d iag   ⋅ ( ζ − ζ ae )     ωi    ωi    

−1

(6.49)

where I is the identity matrix ( N m od by N m od ), and where ζ , ζ ae and κ ae are defined

(

)

in Eq. 4.68. By introducing the modal stiffness matrix K 0−1 = d iag 1 / ωi2 M i  , the 

definition of m i in Eq. 6.41 and the notation in Eqs. 5.24 and 5.25, then the content of


κ ae

%  κ aeij =  $

$    %

are given by

κ aeij =

K aeij

ωi2 M i

=

ρB

ζ ae

and

2

2m i

∫

⋅

L exp

(φ

$    %

(6.50)

)

∫ ( φi

(6.51)

)

T

⋅ φi d x

L

ζ aeij

%  ζ aeij =  $

⋅ Kˆ ae ⋅ φ j d x

T i

∫ ωi C aeij ρ B 2 L exp = = ⋅ 2 ωi2 M i 4 m i

131

(φ

)

ˆ ⋅φ dx ⋅C ae j

T i

∫ ( φi

T

(6.52)

)

⋅ φi d x

L

Fully expanded versions of these expressions are given by

κ aeij ρB 2 2m i

 =  ∫ φ yi φ y j P4* + φz i φ y j H 6* + φθi φ y j B A 6* + φ yi φz j P6* + φz i φz j H 4* + φθi φz j B A 4* L  exp

(

  2 2 2  ∫ φ yi + φz i + φθi d x    L (6.53)

)

(

+φ yi φθ j B P3* + φz i φθ j B H 3* +φθi φθ j B 2 A 3* d x  

ζ aeij ρB 2 4 m i

)

 =  ∫ φ yi φ y j P1* + φz i φ y j H 5* + φθi φ y j B A 5* + φ yi φz j P5* + φz i φz j H 1* + φθi φz j B A1* L  exp

(

)

+φ yi φθ j B P2* + φzi φθ j B H 2* +φθi φθ j B 2 A 2* d x  

  2 2 2  ∫ φ yi + φz i + φθi d x   L 

(

)

(6.54) As mentioned above, the normalised modal load matrix SQˆ ( N m od by N m od ) is given in Eq. 4.75. Its content S Qˆ Qˆ (ω ) , containing the cross sectional load matrix Sqq ( ∆x , ω ) , is i j

defined in Eq. 4.77 (and 4.78). Based on the buffeting load expressions in chapter 5.1 it is now only Sqq that remains for further expansion. Recalling from Eq. 6.32 that the


132

T

buffeting part of the cross sectional loading is q y q z qθ  = Bq ⋅ v = ( ρVB / 2 ) ⋅ Bˆ q ⋅ v , then its Fourier transform is

a q y    ˆ ⋅a aq ( x , ω ) =  a q z  = ( ρVB / 2 ) ⋅ B q v    a qθ 

av ( x , ω ) = [ a u

where:

(6.55)

aw ]

T

(6.56)

The cross spectrum Sqq ( ∆x , ω ) is then given by Sqq ( ∆x , ω ) = lim

1  * T   aq ( x1 , ω ) ⋅ a q ( x 2 , ω ) 

T →∞ π T

2

1  *  ρVB  ˆ ˆT = av ( x1 , ω ) ⋅ aTv ( x 2 , ω )  ⋅ B q  ⋅ Bq ⋅ Tlim  →∞ π T 2  

(6.57)

2

 ρVB  ˆ ˆT =  ⋅ Bq ⋅ Sv ( ∆x , ω ) ⋅ Bq 2   where Sv ( ∆x , ω ) is defined in Eq. 6.37. Adopting the assumption that S u w = S w u ≈ 0 ,

see Eq. 6.17, and introducing Eq. 6.40, then the content of the normalised modal load matrix (Nmod by Nmod) % $    S Qˆ Qˆ (ω ) SQˆ (ω ) =  i j   $ % is given by

S Qˆ Qˆ

i j

2

∫∫

 ρV B  L exp (ω ) =   ⋅  2  2

{

}

ˆ ( ∆x , ω )  ⋅ B ˆ ⋅ I2 ⋅ S ˆ T ⋅ φ (x )d x d x φTi ( x1 ) ⋅ B q v v j 2 1 2  q

(ω M ) ⋅ (ω M ) 2 i

2

=

(6.58)

ρB 3 ρB 3  V   V ⋅ ⋅  ⋅ 2m i 2m j  B ωi   B ω j

i

2 j

j

2

 2  ⋅ Jˆ ij   (6.59)

Thus, in case of multi mode calculations there will be N m od ⋅ N m od such reduced joint acceptance functions Jˆ ij2 , each defined by


∫∫

Jˆ ij2

=

L exp

{

133

}

ˆ ( ∆x , ω )  ⋅ B ˆ ⋅ I2 ⋅ S ˆ T ⋅ φ (x )d x d x φTi ( x1 ) ⋅ B q v v j 2 1 2  q (6.60)

 T     ∫ φi ⋅ φi d x  ⋅  ∫ φTj ⋅ φ j d x      L  L 

A fully expanded version of J ij2 is given by

J ij2 =

2 2   D  2ˆ D  2ˆ  ′ + − x x C I S C C φ φ 2  ( ) ( )  ∫∫ yi 1 y j 2  B D  u u u  B D L  I w S w w   L exp   

2   2 D   +φz i ( x1 ) φz j ( x 2 )  2C L I u2 Sˆ u u +  C L′ + C D  I w2 Sˆw w  B     2 2 ′ ) I w2 Sˆw w  +φθi ( x1 ) φθ j ( x 2 )  2 B C M I u2 Sˆ u u + ( B C M    D  D D   + φ yi ( x1 ) φz j ( x 2 ) + φz i ( x1 ) φ y j ( x 2 )  4 C D C L I u2 Sˆ u u +  C D′ − C L   C L′ + C D  I w2 Sˆw w    B B B   

(

)

(

)

 D  D  ′ I w2 Sˆw w  + φ yi ( x1 ) φθ j ( x 2 ) + φθi ( x1 ) φ y j ( x 2 )  4 C D B C M I u2 Sˆ u u +  C D′ − C L  B C M   B B      D   ′ I w2 Sˆw w   d x1d x 2 + φz i ( x1 ) φθ j ( x 2 ) + φθi ( x1 ) φz j ( x 2 )  4 C L B C M I u2 Sˆ u u +  C L′ + C D  B C M   B    

(6.61) and the corresponding reduced version is given by Jˆ ij2 =

J ij2      ∫ φ y2i + φz2i + φθ2i d x  ⋅  ∫ φ y2j + φz2j + φθ2j d x      L  L 

(

)

(

)

(6.62)

The reduced cross spectra Sˆ u u and Sˆw w are defined by Sˆ u u = S u u ( ∆x , ω ) / σ u2   Sˆw w = S w w ( ∆x , ω ) / σ w2 

(6.63)

where S u u and S w w are defined in Eq. 3.39. (A transition between spectral density descriptions using f rather than ω as the frequency variable is shown in Eq. 2.68.) Since spatial averaging will eliminate any complex parts of the cross spectra, Eq. 6.63 may for all practical purposes be replaced by

134


S (ω ) ˆ Sˆ u u = Re S u u ( ∆x , ω )  / σ u2 = u 2 ⋅ Co u u ( ∆x , ω )

      

σu

Sˆw w = Re S w w ( ∆x , ω )

 / σ w2

S (ω ) ˆ = w 2 ⋅ Co w w ( ∆x , ω )

σw

(6.64)

ˆ ˆ where Co u u and Cow w are the reduced u- and w- component co-spectra (see Eq. 3.40).

Example 6.3: Let us again (similar to example 6.2) consider a simply supported horizontal beam type of bridge with span L = 500 m that is elevated at a position z f = 50 m , but now we set out to calculate the dynamic response at x r = L 2 associated with the two mode shapes φ1 = 0 φz1

0 

T

φ2 = 0 0 φθ2 

and

T

with corresponding eigen–frequencies ω1 = 0.8 and ω2 = 2.0 rad / s . As can be seen, φ

1

contains only the displacement component in the across wind vertical direction while φ2 only contains torsion. Let us for simplicity assume that φz1 = φθ2 = sin π x L . Thus, the aim of this

example is to calculate the corresponding dynamic response quantities σ rz rz and σ rθ rθ

at

x r = L 2 and the covariance Cov rθ rz between them. It is taken for granted that the chosen mean wind velocity settings are well below any instability limit, such that any changes to resonance frequencies may be ignored. Again, it is assumed that the cross section is close to a flat plate with the following static load coefficient properties: D ′ = 1.5 CL = 0 C L′ = 5 CM = 0 CM and C D C L′ B (Quantifying the drag coefficient is obsolete since y direction response is not excited.) Let us also assume that the entire span is flow exposed, i.e. L exp = L , and adopt the following wind field properties: 1) the turbulence intensity 2) the integral length scales:

3) the auto spectral density:

I w = σ w V = 0.08 xf

 zf  L u = 100 ⋅    10 

S w (ω )

σ w2

4) the normalised co-spectrum: where Cw x = Cw y f = 6.5 / ( 2π ) ≈ 1.0 .

=

0.3

1.5 ⋅

= 162m , xf

(1 + 2.25 ⋅ ω ⋅

Lw V xf

Lw V

(see Eq. 3.14) xf

Lw =

xf

)

53

ω ⋅ ∆x   ˆ Co w w ( ω , ∆x ) = exp  −Cw x ⋅ V  

Lu (see Eq. 3.36), 12 (see Eq. 3.25)

(see Eq. 3.41)


135

Let us allot the following values to the remaining constants that are necessary for a numerical calculation of the relevant dynamic response quantities at x r = L 2 :

ρ (kg/m3)

B (m)

D (m)

1.25

20

4

m1

2

(kgm /m)

(kg/m) 10

ω1

ω2

ζ1

ζ2

(rad/s) 0.8

(rad/s) 2.0

0.005

0.005

m2

4

6 ⋅ 10

5

Since m 1 and m 2 are constant along the span, then the modally equivalent and evenly distributed masses m 1 = m 1 and m 2 = m 2 . It should be noted that φ1T ⋅ φ1 = φz21 = sin 2 π x L L

and that

L

∫ φm ⋅ φn d x = 2

φT2 ⋅ φ2 = φθ22 = sin 2 π x L

and

m  = z1 or θ2 . n

for any combination of

0

Finally, let us for simplicity adopt quasi-static values to the aerodynamic derivatives, except for ′ (V B ωi ) A 2* which is responsible for aerodynamic damping in torsion. Adopting A 2* = − β M C M

2

and β M = 0.2 provides a good approximation to the flat plate properties. Thus, the aerodynamic derivatives associated with motion in the across wind vertical direction and torsion are given by (see Eq. 5.26):  −Vˆ   H 1*   H 4* A 4*   A1*   Vˆ     *    *   ′ ⋅  − β M ⋅ Vˆ 2   H 2  = −C L′ ⋅  0   A2  = CM  H 5* A 5*  = 0    *  *  *  ˆ2 * 2  Vˆ   H 3   A 3   H 6 A 6  V  where: Vˆ = V ( B ω ) . The aerodynamic coefficients associated with changes in stiffness and i

damping are then given by (see Eq. 6.51 and 6.52, or the fully expanded versions in Eqs. 6.53 and 6.54):

κ aeij = ζ aeij =

ρB 2 2m i

ρB 2 4 m i

⋅

∫

(φ φ

∫

(φ φ

L exp

⋅

L exp

* zi θ j B H 3

* zi z j H 1

)

+ φθi φθ j B 2 A 3* d x

∫ (φzi + φθi ) d x 2

2

L

)

+ φθi φz j B A1* + φθi φθ j B 2 A 2* d x

∫ (φzi + φθi ) d x 2

2

L

where in this case i and j are equal to 1 or 2. Introducing the choice of aerodynamic derivatives given above, then:

κ ae11 = 0 ,

κ ae12 =

ρB

2

2m 1

∫

⋅

L exp

φz1 φθ2 B H 3*d x

∫ φz1 d x 2

L

=

ρB 3 2m 1

⋅ H 3* =

 V  ⋅ C L′ ⋅   2m 1  B ω1 

ρB 3

2

136


κ ae21 = 0 ,

κ ae22 =

ρB

2

2m 2

∫

⋅

L exp

φθ22 B 2 A 3*d x =

∫ φθ2 d x 2

ρB 4 2m 2

⋅ A 3* =

 V  ′ ⋅ ⋅CM  2m 2  B ω2 

ρB 4

2

L

ζ ae11 =

ρB

2

4 m 1

∫

⋅

L exp

∫

L

ζ ae21 =

ρB

2

4 m 2

∫

⋅

L exp

φz21 H 1*d x φz21 d x

∫

ζ ae22 =

4 m 2

∫

⋅

L exp

ρB 2 4 m 1

φθ2 φz1 B A1*d x

L

ρB 2

=

=

φθ22 d x

φθ22 B 2 A 2*d x

∫ φθ2 d x 2

=

⋅ H 1* = −

ρB 3 4 m 2

ρB 4 4 m 2

ρB 2 4 m 1

⋅ A1* = −

⋅ A 2* = −

⋅ C L′ ⋅

ρB 3 4 m 2

V B ω1

′ ⋅ ⋅CM

ζ ae12 = 0

,

V B ω2

 V  ′ ⋅ ⋅ βM CM  4 m 2  B ω2 

ρB 4

2

L

The non-dimensional frequency response function is then given by (see Eq. 6.49) 2     1  1    ˆ Hη (ω ) = I − κ ae −  ω ⋅ d iag    + 2iω ⋅ d iag   ⋅ ( ζ − ζ ae )    ω ω  i   i   

 1 0  0 κ ae  ω −2 12  − ω2  1  −  0  0 1  0 κ ae22 

where:

−1

ω −1 0  0   ζ 1 0  ζ ae11 + 2iω  1   − ω2−2   0 ω2−1    0 ζ 2  ζ ae21

κ ae12 = 9.766 ⋅ 10 −4 ⋅ V 2 ,

κ ae22 = 1.563 ⋅ 10 −4 ⋅ V 2 ,

= 0      ζ ae22    

−1

ζ ae11 = −39.06 ⋅10 −4 ⋅ V ,

ζ ae21 = −15.63 ⋅ 10 −4 ⋅ V , ζ ae22 = −0.1563 ⋅ 10 −4 ⋅ V 2 , and where all other quantities are given above. The aerodynamic stiffness and damping coefficients κ ae12 , κ ae21 , ζ ae11 , ζ ae21 , ζ ae22 are shown in Fig. 6.8. The absolute value of the determinant of the non–dimensional frequency response function (at V = 0 ) is shown in Fig. 6. 9 together with the single point spectral density and normalised co-spectrum of the wind turbulence w component.


Fig. 6.8

137

Aerodynamic stiffness and damping coefficients

Fig. 6.9 Top left and right hand side diagrams: w component spectral density and normalised co-spectrum, lower left: absolute value of the determinant of the nondimensional frequency response function at V = 0 , lower right: the joint acceptance function of normalized mode shapes.

138


The content of the normalised modal load matrix  S Qˆ Qˆ 1 1 SQˆ (ω ) =  S Qˆ Qˆ  2 1

  S Qˆ Qˆ  2 2 S Qˆ

ˆ 1 Q2

is given in Eq. 6.59: 2

2

S Qˆ Qˆ

i j

ρB 3 ρB 3  V   V  ˆ2  ⋅ J ij (ω ) = ⋅ ⋅   ⋅ 2m i 2m j  B ωi   B ω j 

where the reduced joint acceptance function Jˆ ij is given in Eq. 6.60. An expanded version of the joint acceptance function itself is given in in Eq. 6.61. Under the present circumstances it simplifies into 2 = J 11

φz1 ( x1 ) ⋅ φz1 ( x 2 ) ⋅ (C L′ I w ) ⋅

∫∫

′ I w2 ⋅ φz1 ( x1 ) ⋅ φθ2 ( x 2 ) ⋅ C L′ B C M

∫∫

′ Iw ) ⋅ φθ2 ( x1 ) ⋅ φθ2 ( x 2 ) ⋅ ( B C M

L exp 2 = J 12

L exp 2 = J 22

S w w (ω , ∆x )

∫∫

2

L exp

σ w2

2

d x1d x 2

S w w (ω , ∆x )

σ w2

2 2 J 21 = J 12

d x1d x 2 ,

S w w (ω , ∆x )

σ w2

d x1 d x 2

ˆ Introducing S w w (ω , ∆x ) = S w (ω ) ⋅ Co w w (ω , ∆x ) and I w = σ w V , then the content of the normalised modal load matrix is given by 2

2

S Qˆ

ˆ 1 Q1

S Qˆ

ˆ 2 Q1

 ρVB C ′  (ω ) =  2 L ⋅ Jˆ11 (ω )  S w (ω ) ,  2ω1 m 1 

S Qˆ

(ω ) = S Qˆ1Qˆ 2 (ω )

S Qˆ

and

ˆ 1 Q2

ˆ 2 Q2

 ρVB C L′ B C M  ′ ⋅ Jˆ 21 (ω )  S w (ω ) (ω ) =    2ω1ω2 m 1m 2   ρVB 2C M ′

(ω ) = 

2  2ω2 m 2

,

2

 ⋅ Jˆ 22 (ω )  S w (ω )  

where: 2 Jˆ11 =

∫∫

 2   ∫ φz1 d x    L 

∫∫

ˆ φz1 ( x1 ) ⋅ φθ2 ( x 2 ) ⋅ Co w w (ω , ∆x ) d x 1 d x 2

 2   ∫ φz1 d x ⋅ ∫ φθ22 d x    L L 

∫∫

ˆ φθ2 ( x1 ) ⋅ φθ2 ( x 2 ) ⋅ Co w w (ω , ∆x ) d x 1 d x 2

 2   ∫ φθ2 d x    L 

L exp 2 Jˆ 21 =

L exp 2 Jˆ 22 =

2

ˆ φz1 ( x1 ) ⋅ φz1 ( x 2 ) ⋅ Co w w (ω , ∆x ) d x 1 d x 2

L exp

(

ˆ Since φz1 = φθ2 = sin π x L , and Co w w ( ω , ∆x ) = exp −Cw y ⋅ ω ⋅ ∆x V

equivalent to that which was encountered in Example 6.2, and thus,

)

2

the present situation is


2  Jˆ11  2  Jˆ 21  = 4 ⋅ψ ( ω ) 2  ˆ J 22  

ψ (ω ) =

where

139

ωˆ 2

ωˆ + π

2

+ 2π 2 ⋅

1 + exp ( −ωˆ )

(ωˆ

2

+π2

)

2

and where ωˆ = C u x ω L exp V . The normalised modal load matrix SQˆ is then given by 2   2 2  ω2  m   ′ ′ C B C L′ C M   S Qˆ Qˆ  ( ρVB )2 ⋅ S (ω ) ⋅ψ (ω )  L ω  S Qˆ Qˆ   1  m 1 1 1 1 2 w   = SQˆ (ω ) =   2 2 2 S Qˆ Qˆ S Qˆ Qˆ  ω1 m 1 ⋅ ω2 m 2  1  2  ω1  m 2 2  2 1 ′  B C L′ C M ( B C M′ )      ω2  m 2  And thus, the spectral density response matrix at x r = L 2 is given by (see Eqs. 4.81 and 4.82)

(

)(

 S rz rz Srr ( L 2 , ω ) =  S rθ rz

)

S rz rθ   = Φr ( L 2 ) ⋅ Sη (ω ) ⋅ ΦTr ( L 2 ) S rθ rθ 

ˆ * (ω ) ⋅ S (ω ) ⋅ H ˆ T (ω ) and Φ ( L 2 ) = 1 0  where: Sη (ω ) = H r η η 0 1  Qˆ   E E (ω ) =  11  E 21

Introducing the impedance matrix 2

ω ω E 11 = 1 −   + 2i ζ 1 − ζ ae11 , ω ω 1  1

E 21 = −2i

ω ζ ω2 ae21

(

)

where

E 12 = −κ ae12 , 2

ω  ω ζ − ζ ae22 E 22 = 1 − κ ae22 −   + 2i ω2 2  ω2 

and

 ˆ ˆ (ω ) =  H 11 H η  Hˆ 21

then

E 12  E 22 

Hˆ 12  1  E 22  = E−1 =  det E  − E 21 Hˆ 22 

(

)

− E 12   E 11 

rendering the following expression for the spectral density response matrix at x r = L 2 Srr ( L 2 , ω ) =

ρB 2 ρB 4  V  ⋅ ⋅  m 1 m 2  B ω1 

2

2  Sˆη  V  2 S w (ω )  11 I ψ ω ⋅ ⋅ ⋅ ⋅ ⋅ ( )  w 2 Sˆ B ω σ 2   w  η21

Sˆη12   Sˆη22 

where: 2

Sˆη11 (ω ) = γ L L ⋅ Hˆ 11 + 2 ⋅ γ L M ⋅ Hˆ 12 ⋅ Hˆ 11 + γ M M ⋅ Hˆ 12

( ⋅ ( Hˆ

2

) )+γ

Sˆη12 (ω ) = γ L L ⋅ Hˆ 11 ⋅ Hˆ 21 + γ L M ⋅ Hˆ 12 ⋅ Hˆ 21 + Hˆ 11 ⋅ Hˆ 22 + γ M M ⋅ Hˆ 12 ⋅ Hˆ 22 Sˆη21 (ω ) = γ L L ⋅ Hˆ 11 ⋅ Hˆ 21 + γ L M 2

21

⋅ Hˆ 12 + Hˆ 22 ⋅ Hˆ 11

Sˆη22 (ω ) = γ L L ⋅ Hˆ 21 + 2 ⋅ γ L M ⋅ Hˆ 21 ⋅ Hˆ 22 + γ M M ⋅ Hˆ 22 2

MM

⋅ Hˆ 22 ⋅ Hˆ 12

2

2

 ω2  m 2 2ω  m ′ )  1  1 = 2.4 . ′ = 150 and γ MM = ( B C M = 9375 , γ L M = B C L′ C M  ω m ω m 2  1 1  2 Since we are mainly aiming at calculating the content of the covariance matrix

γ L L = C L′2 

140


∞  σ r2 r z z Covrr ( x r = L 2 ) = ∫ Srr ( L 2 , ω ) d ω =   Cov 0 rθ rz  it is only the absolute values that are of interest.

Cov rz rθ   σ r2θ rθ 

Fig. 6.10 Top left: absolute value of frequency response function. Top right: cross spectrum between vertical and torsion response components. Lower left and right: spectra of components in vertical direction and torsion. V = 30 m s . The absolute value of the determinant of the non-dimensional frequency at a mean wind velocity of V = 30 m s is shown in the top left hand side diagram in Fig. 6.10. The top right hand side diagram shows the amplitude of the cross spectrum between rz and rθ while the two lower diagrams show the spectral densities of rz and rθ , all at a mean wind velocity of V = 30 m s . As can be seen, there are traces of modal coupling. In this case the coupling effects are exclusively motion induced. Comparing det H (ω ) shown in the top left hand side diagram of Fig. 6.10 to

that which is shown in Fig. 6.9 it is seen that the resonance frequency associated with the second mode shape (in torsion) is no longer precisely at 2 rad s , but slightly below. It is also seen that the resonance peaks are reduced, and particularly the peak associated with φz1 at ω1 = 0.8 rad/s. The standard deviation of the dynamic responses in the across wind direction ( rz ) and in torsion ( rθ ) at various mean wind velocities are shown on the two left hand side diagrams in Fig. 6.11. The circular points joined with a fully drawn line are based on the development shown above, i.e. they contain the effects of aerodynamic derivatives, while the broken line represents the situation that aerodynamic derivatives are ignored. As can be seen, the difference is considerable for the


141

respone in the across wind vertical direction, but in torsion only at the highest mean wind velocity setting. It should be noted that the applicability of quasi static aerodynamic derivatives is in many cases questionable, and they should in general be replaced by values obtained from wind tunnel tests. The covariance coefficient between the dynamic responses rz and rθ is shown on the top right hand side diagram in Fig. 6.11, and again, circles and fully drawn line contain the effects of aerodynamic derivatives while for the broken line no motion induced effects have been included. The changes of the resonance frequency associated with the second mode shape (in torsion) at increasing mean wind velocities is shown on the lower right hand side diagram in Fig. 6.11. As can be seen, the reduction of the resonance frequency from V = 0 to V = 40 m s is slightly less than 15 % (which without further iterations implies an overestimation of the torsion response).

Fig. 6.11 Top and lower left: dynamic response in vertical direction and torsion. Top right: covariance coefficient. Lower right: resonance frequency associated with 2nd mode. Full lines: including motion induced effects. Broken lines: without motion induced effects.

142

6.4


Vortex shedding

As shown in chapter 5.3, the vortex shedding induced load effects at or in the vicinity of lock-in are dependent on the dynamic response of the structure, i.e. the total damping in each mode is unknown prior to any knowledge about the actual structural displacements. Thus, the calculation of vortex shedding induced dynamic response will inevitably involve iterations. It should be acknowledged that the peak factor for vortex shedding response does not comply with the theory behind what may be obtained from Eq. 2.45. For an ultranarrow-banded vortex shedding response the peak factor is close to 1.5 (theoretically 2 , see Eq. 2.47). For broad-banded response Eq. 2.45 will most often render conservative results. Some time domain simulations of response spectra (see Appendix A) will give a good indication on what peak factor should be chosen. Multi mode response calculations The general solution of a multi mode approach to the problem of calculating vortex shedding induced dynamic response is identical to that which has been presented above for buffeting response calculations. I.e., the general solution to the calculation of the three by three cross spectra response matrix Srr ( x r , ω ) is given in Eq. 4.80–4.82, while the corresponding covariance matrix is given in Eqs. 6.47 and 6.48. The N m od by N m od frequency response matrix Hˆ η (ω ) and the modal load matrix SQˆ (ω ) are given in Eqs. 4.69 and 4.75, except that for vortex shedding the motion induced load is assumed exclusively related to structural velocity, and its effect applies to the actual modal response and not to the individual Fourier components. As shown in Eq. 5.36, this

(

)

implies that K ae = 0 and Cae = ρ B 2 / 2 ⋅ ωi (V ) ⋅ d iag 0 H 1* B 2 A 2*  , and thus 2     ˆ (ω ) = I −  ω ⋅ d iag  1   + 2iω ⋅ d iag  1  ⋅ ( ζ − ζ )  H     η ae   ωi    ωi    

where ζ = d iag [ζ i ] and the content of ζ ae is given by

−1

(6.65)

6.4 VORTEX SHEDDING

ζ aeij

∫ ωi C aeij ρ B 2 L exp = = ⋅ 2 ωi2 M i 4 m i

(φ

)


T i

∫ ( φi

T

)

⋅ φi d x

L

=

ρB 2 4 m i

∫

⋅

L exp

143

(6.66)

φiz φ jz H 1*d x + B 2 ∫ φiθ φ jθ A 2*d x

∫ (φyi 2

L

)

+ φz2i + φθ2i d x

where H 1* and A 2* , are given in Eq. 5.37 and where m i is defined in Eq. 6.41. If H 1* and A 2* are taken as modal constants and independent of span-wise position, then ζ ae

becomes diagonal due to the orthogonal properties of the mode shapes, i.e. ζ ae = d iag ζ aei  where

ζ aei =

ρB

2

4 m i

H 1* ⋅

∫

L exp

φi2z d x + B 2 A 2* ∫ φi2θ d x

∫ (φ

L

(6.67)

2 iy

(6.68)

)

+ φi2z + φi2θ d x

This implies that Hˆ η (ω ) is an N m od by N m od diagonal matrix. In vortex shedding induced vibration problems it is usually not essential to include the along wind load effects. The load vector may then be reduced to

q ( x , t ) = [0 q z

qθ ]

T

(6.69)

and the corresponding Fourier transform is aq ( x , ω ) = 0 a q z

a qθ 

T

(6.70)

The cross sectional load spectrum is defined by (see Eq. 4.78) 0 0  1 1 * T *  0 a qz a qz Sqq ( ∆x , ω ) = lim aq aq = lim T →∞ π T T →∞ π T  0 a * a qθ q z 

(

)

0 a q*z a qθ a q*θ a qθ

  0  0  = 0 S q q z z    0 S q q θ z  

0 S q z qθ S qθ qθ (6.71)

    


144

The problem is greatly simplified if the cross coupling between q z and qθ may be disregarded, in which case

0 0  Sqq ( ∆x , ω ) ≈ 0 S q z q z  0 0

0 0 S qθ qθ

    

(6.72)

where the cross spectra S q z q z and S qθ qθ are given by

ˆ  S q z q z = S q z (ω ) ⋅ Co q z ( ∆x )   ˆ S qθ qθ = S qθ (ω ) ⋅ Co qθ ( ∆x )  

(6.73)

The single point spectra S q z and S qθ are defined in Eq. 5.33, while the reduced co– ˆ ˆ spectra Co q z and Coqθ are defined in Eq. 5.34. Thus, the elements of SQˆ (see Eqs. 4.75

– 4.78) are reduced to

S Qˆ Qˆ (ω ) =

∫∫

φTi ( x1 ) ⋅ Sqq ( ∆x , ω ) ⋅ φ j ( x 2 ) d x1 d x 2

L exp

∫∫ {φiz ( x1 ) φ jz ( x 2 ) S qz qz

= S qz =

∫∫

L exp

L exp

(ω M ) ⋅ (ω M ) 2 i

i j

2 j

i

j

}

+ φiθ ( x1 ) φ jθ ( x 2 ) S qθ qθ d x1 d x 2

(ω M ) ⋅ (ω M ) 2 i

2 j

j

ˆ dx dx + S φiz ( x1 ) φ jz ( x 2 ) Co qz qθ 1 2

∫∫

(

i

ωi2 M i

)⋅(

L exp

ω 2j M j

ˆ dx dx φiθ ( x1 ) φ jθ ( x 2 ) Co qθ 1 2

) (6.74)

Furthermore, it is a reasonable assumption that the integral length–scale of the vortices λ D is small as compared to the flow exposed length L exp of the structure, and since q z and qθ are caused by the same vortices their coherence properties are likely to be identical, in which case (see procedure presented in example 6.1)

6.4 VORTEX SHEDDING

S Qˆ Qˆ

i j

 2 λ D S q z   (ω ) ≈

∫

L exp

φiz ( x ) φ jz ( x ) d x + S qθ

∫

145 

L exp

φiθ ( x ) φ jθ ( x ) d x   

(ω M ) ⋅ (ω M ) 2 i

2 j

i

(6.75)

j

Again, due to the orthogonal properties of the mode shapes this implies that SQˆ becomes diagonal, i.e. SQˆ = d iag S Qˆ   i

(6.76)

where

S Qˆ

i

  2 λ D S q z (ω ) ∫ φi2z d x + S qθ (ω ) ∫ φi2θ d x    L exp L exp   (ω ) = 2 ωi2 M i

(

(6.77)

)

The calculation of the spectral response matrix is given in Eqs. 4.80 – 4.82, though, it should be noted that if the simplifications above hold then both Hˆ η and SQˆ are diagonal, in which case Srr ( x r , ω ) = Φr ( x r ) ⋅ d iag S η (ω )  ⋅ ΦTr ( x r ) =  i 

=

N m od

∑ i =1

N m od

∑ i =1

φi ( x r ) ⋅ φTi ( x r ) ⋅ S η (ω ) i

φ y2 ( x r ) φ y ( x r ) ⋅ φz ( x r ) φ y ( x r ) ⋅ φθ ( x r )     φz2 ( x r ) φz ( x r ) ⋅ φθ ( x r )  ⋅ S ηi (ω )   φθ2 ( x r )  S ym .   i (6.78)

where

2

S η (ω ) = Hˆ η (ω ) ⋅ S ˆ (ω ) i i Q

(6.79)

i

Hˆ η is given by (see Eq. 6.65) i

  ω 2 ω Hˆ η (ω ) = 1 −   + 2i ⋅ ζ i − ζ aei ⋅  i ωi    ωi   

(

)

−1

(6.80)

and ζ aei is given in Eq. 6.68 (see also 5.37). The corresponding covariance response matrix Covrr ( x r ) for the dynamic response at span-wise position x r is then given by


146

∞

Covrr ( x r ) = ∫ Srr ( x r , ω ) d ω 0

 σ r2 r  yy

Cov ry rθ   Cov rz rθ  =  σ r2θ rθ  

Cov ry rz

=  S ym . 

σ r2z rz

N m od

∑ i =1

φ y2 ( x r ) φ y ( x r ) ⋅ φz ( x r ) φ y ( x r ) ⋅ φθ ( x r )     φz2 ( x r ) φz ( x r ) ⋅ φθ ( x r )  ση2i   φθ2 ( x r )  S ym .   i (6.81)

∞

ση2i = ∫ S ηi d ω

where

(6.82)

0

is the variance contribution from an arbitrary mode i . Usually, vortex shedding induced dynamic response is largely resonant and narrow-banded. It will then usually suffice to only consider the resonant part of the frequency domain integration in Eq. 6.82, and discard the background part. Thus, ∞ ∞ ∞ πωi ⋅ S Qˆ (ωi ) 2 2 i ση2i = ∫ S ηi d ω = ∫ Hˆ ηi (ω ) ⋅ S ˆ (ω ) d ω ≈ ∫ Hˆ ηi (ω ) d ω ⋅ S ˆ (ωi ) = Qi Qi 4 ζ − ζ 0 0 0 i aei

(

)

(6.83) where (see Eqs. 6.77 and 5.33)

S

Qˆ i

  2 λ D S q z (ωi ) ∫ φi2z d x + S qθ (ωi ) ∫ φi2θ d x    L exp L exp   (ωi ) = 2 2 ω M

(

=

2λ D

(ω M ) ( Bσ θ ) +

2

2 i

( ρV ⋅

i

2

q

bθ

∫

L exp

2

B /2

π ⋅ ωs

i

i

)

2

)

 2 σ q z ⋅  bz

∫

L exp

φi2z d x

  1 − ω / ω 2  i s ⋅ exp  −    bz      

  1 − ω / ω 2   i s   bθ       

φi2θ d x ⋅ exp  − 

(6.84) and ωs = 2π fs . As mentioned above, the calculations will inevitably demand iterations, because H 1* and A 2* are functions of σ rz rz and σ rθ rθ . The iteration will take place on the difference between ζ i and ζ aei , which in general will be a small quantity.

6.4 VORTEX SHEDDING

147

Example 6.4: Let us consider a simply supported horizontal beam type of bridge with span L = L exp = 500 m and set out to calculate the vortex shedding induced dynamic response at x r = L 2 which is associated with the three mode shapes

0  0      πx φ1 = φz1  = sin    L    0  0 

      

0   0       3π x φ2 = φz 2  = sin    L     0  0 

      

and

 0   0    0 φ3 =  0  =  φ     θ3  sin π x  L  

       

with corresponding eigen-frequencies 0.8 , 1.6 and 2.5 rad/s. As can be seen, φ1 and φ2 contain only the displacement component in the across wind vertical direction while φ3 only contains torsion. Let us adopt the following structural properties:

ρ kg

B

D

mz

m3

m

m

kg m

1.25

20

4

4

10

mθ k gm m

2

6 ⋅ 10

5

ω1 = ωz1

ω2 = ωz 2

ω3 = ωθ3

ζ1 = ζ 3

ζ2

rad s

rad s

rad s

%

%

0.8

1.6

2.5

0.5

0.75

and the following vortex induced wind load properties: St

σˆ q z

σˆ qθ

bz

bθ

az

aθ

λz = λθ

K az 0

K a θ0

0.1

0.9

0.3

0.15

0.1

0.4

0.1

1.2

0.2

0.02

where:

1



σˆ q z = σ q z  ρV 2 B  2 

and

1



σˆ qθ = σ qθ  ρV 2 B 2  2 

Since m z and m θ are constant along the span, then the modally equivalent and evenly distributed masses m 1 = m 2 = m z and m 3 = m θ .


148

Fig. 6.12 Aerodynamic damping coefficient Finally, let us adopt the following wind velocity variation of the relative aerodynamic damping coefficient (see Fig. 6.12) −n −m  ω D  ⋅ exp  − V V R i K a (V ) K a 0 = 2.6 ⋅ V V R i V Ri = i ⋅ where  2 π St   In this case (see Eq. 6.81)

(

)

(

 σ r2 r Cov r r y z  yy Covrr ( x r ) = ∫ Srr ( x r , ω ) d ω =  σ r2z rz  0 S ym .  ∞

)

Cov ry rθ   Cov rz rθ   2 σ rθ rθ  

0 0 0 0 0 0 0 0 0     2   2  2 2 = 0 φz1 ( x r ) 0  ⋅ ση1 + 0 φz 2 ( x r ) 0  ⋅ ση2 + 0 0 0  ⋅ ση23       2 0 0 0 0 0 0 φθ3 ( x r )  0 0

and thus: 0 0  Covrr ( x r ) = 0 σ r2z rz  0 0 

0  0 0   0  = 0 φz21 ( x r ) ⋅ ση21 + φz22 ( x r ) ⋅ ση22   0 σ r2θ rθ  0

   0  φθ23 ( x r ) ⋅ ση23  0

From Eqs. 6.83 and 6.84 (and taking it for granted that λz = λθ = λ ) the following variance contributions are obtained

6.4 VORTEX SHEDDING

ση21

D ∫ φ12z d x 2 ˆ σ   L exp 1 D ρ B D qz λ = 72 74 ⋅ ⋅ 2 ⋅ ⋅ ⋅ ⋅ g12 V R1 , V 2 2 π  b ζ −ζ m S t z ae1 1 1  2     ∫ φ1 z d x    L 

(



ση22 = 

D

7 2 7 4 2 π

ση23

149

)

D ∫ φ22z d x 2 ˆ L exp 1 ρ B D σ qz  λ ⋅ ⋅ ⋅ ⋅ g 22 V R 2 , V  ⋅ ⋅ 2 m 2 S t 2  bz ζ 2 − ζ ae2    ∫ φ22z d x    L 

(

)

D ∫ φ32θ d x 2 2   ρ ( B D ) σˆ qθ L exp 1 1 λ = 72 74 ⋅ ⋅ 2 ⋅ ⋅ ⋅ ⋅ g 32 V R 3 , V 2 2 π  m 3 b − ζ ζ St 3 ae θ   3    ∫ φ32θ d x    L 

(

)

where 3 2

 V g1 V R1 , V =    V R1

   

 V g2 V R2 ,V =    V R2

   

 V g 3 V R3 , V =    V R3

   

(

(

(

)

)

)

3 2

32

 1  1 − V R1 / V ⋅ exp  −   2  bz 

  

2

 1  1 − V R2 / V ⋅ exp  −   2  bz 

  

 1  1 − V R3 / V ⋅ exp  −   2  bθ 

  

  

2

  

2

  

            

where

V Ri

ω D = i ⋅ 2π S t

1  i = 2 3 

What then remains are the aerodynamic damping contributions given in Eq. 6.68, from which the following is obtained:

ζ ae1 (V ) =

ρB

2

4 m 1

∫

⋅ H 1* ⋅

L exp

φz21 d x

∫ φz1 d x 2

=

L

ζ ae2 (V ) =

ρB 2 4 m 2

∫

⋅ H 1*

⋅

L exp

φz22 d x

2 ∫ φz 2 d x

=

 σ r r (V  ⋅ K a z V R1 , V ⋅ 1 −  z z 4 m 1   a z D

)

 σ r r (V  V R 2 , V ⋅ 1 −  z z   a z D

)

ρB 2

ρB 2 4 m 2

(

⋅ K az

(

L

ζ ae3 (V ) =

ρB

4

4 m 3

∫

⋅

A 2*

⋅

L exp

φθ23 d x

∫ φθ3 d x 2

L

=

ρB 2 4 m 3

⋅ K aθ

)

(

)

2       2

     

2  σ rθ rθ (V )    V R 3 , V ⋅ 1 −     aθ    

)

The relevant response diagrams are shown in Figs. 6.13 and 6.14 below.

150


Fig. 6.13

Fig. 6.14

Vortex shedding induced across wind response

Vortex shedding induced torsion response

6.4 VORTEX SHEDDING

151

Single mode single component response calculations A single mode single component response calculation is in the following only considered relevant for displacements in the z direction and in torsion. Thus, it is only mode shapes that primarily contain either z or θ components that are relevant. I.e., it is taken for granted that any of the following two conditions apply

φi ( x ) ≈ [0 φz

0 ]i

   

T

φi ( x ) ≈ [0 0 φθ ]i

T

(6.85)

Off diagonal terms in Eq. 6.78 will then vanish, rendering all covariance quantities obsolete, and Srr will simply contain the response variances of the excitation of each mode on its diagonal. Thus, the response spectrum and the displacement variance associated with the excitation of an arbitrary mode i are given by 2

S rn (ω ) = φn2 ( x r ) ⋅ Hˆ ηn (ω ) ⋅ S Qˆ

n

     

(ω )

∞

σ r2n = ∫ S rn (ω ) d ω 0

z n = θ

(6.86)

where

    (6.87) 2  S qn (ω ) ⋅ ∫ φn ( x ) d x  L exp  S Qˆ (ω ) = 2 λ D ⋅ 2  n 2 ωn M n  and where aerodynamic damping properties may be extracted from Eq. 6.68, rendering Hˆ ηn

  ω 2 ω   (ω ) = 1 −   + 2i ⋅ ζ n − ζ aen ⋅  ωn    ωn   

(

(

ζ aez

C aezz = 2ω M z

= z

ρB

2

H 1*

4 m z

∫

⋅

∫

)

φz2 d x

L exp

φz2 d x

)

=

ρB

2

4 m z

⋅ K az

L

ζ aeθ

C aeθθ = 2ω M θ

= θ

ρB

4

A 2*

4 m θ

∫

⋅

φθ2 d x

L exp

∫ φθ d x 2

L

−1

=

ρB 4 4 m θ

⋅ K aθ

φz2 d x     σ 2  L ∫  1 −  z   ⋅ exp   a z D   ∫ φz2 d x     L (6.88)  2 φ d x  θ   σ 2  L ∫  1 −  θ   ⋅ exp  2   aθ   ∫ φθ d x     L

Usually, vortex shedding induced dynamic response is largely resonant and narrowbanded. It will then suffice to only consider the resonant part of the frequency domain integration in Eq. 6.82, and discard the background part. Thus,


152

∞

∞

0

0

2

σ r2n = ∫ S rn d ω ≈ φn2 ( x r ) ⋅ ∫ Hˆ ηn (ω ) d ω ⋅ S Qˆ ⇒ σ r2n = φn2 ( x r ) ⋅

πωn S Qˆ

(

n

n

(ωn )

(ωn )

4 ζ n − ζ aen

z n = θ

)

(6.89)

As mentioned above, it is also a reasonable assumption that the integral length–scale λ D for q z and qθ are identical. Adopting the convenient notation (see Eq. 6.41)

z n = θ

M n = m n ∫ φn2 d x L

(6.90)

and introducing S Qˆ and S qn from Eqs. 6.87 and 5.33, then the following is obtained n

12

σ rz D

=

φz ( x r ) 27 2 π 7 / 4

⋅

 ρ B D σˆ q z 

m z

⋅

1/2

λ

⋅ S t 2  bz ⋅ ζ z − ζ ae z 

(

  2 D  φ d x z ∫   L exp   ⋅ 2 d x φ ∫ z

)

   

(

⋅ g z V R z ,V

)

L

(6.91) 12

σ rθ =

φθ ( x r ) 27 2 π 7 / 4

ρ (BD )

2

⋅

m θ

⋅

σˆ qθ 

λ

⋅ S t 2  bθ ⋅ ζ θ − ζ ae θ 

(

1/2

)

   

  2 D φ d x ∫ θ   L exp  ⋅ 2 ∫ φθ d x

(

⋅ gθ V Rθ , V

)

L

(6.92) where  V g n V R n ,V =   VR  n

(

)

3/2

   

 1 − V /V 1 Rn ⋅ exp  −   2  bn 

  

2

  

z n = θ

(6.93)

and where V R n = Dωn / ( 2π ⋅ S t ) .

Example 6.5: For a simple beam type of bridge let us set out to calculate the vortex shedding induced dynamic response at x r = L 2 associated with the mode shape φ = [0 φz

0]

T

with corresponding

eigen-frequency ωz = 0.8 rad / s . Let us again for simplicity assume that φz = sin π x L . Thus,

6.4 VORTEX SHEDDING

153

in this case it is only the across wind vertical direction that is of any interest. Typical variation of some basic data is illustrated in Fig. 6.15.

Fig. 6.15 Top left and right: Non–dimensional cross sectional load spectrum and co– spectrum, lower left: aerodynamic damping coefficient, lower right: maximum vortex shedding induced dynamic response vs. ζ z The top left hand side diagram shows the non–dimensional cross sectional load spectrum associated with vortex shedding in the across wind direction (see Eq. 5.33)

ω S q z (ω ) σ q2z

=

1

π bz

 1 − ω ω   ω s ⋅ exp  −    ωs   bz   2

⋅

  1 2 where σ q z = ρV B σˆ q z . The load spectrum is shown for various relevant values of bz , which 2 is the parameter that controls the narrow-bandedness of the process. The reduced co-spectrum (see Eq. 5.34)   ∆x 2   2 ∆x  ˆ −  ∆ = ⋅ Co x cos exp ( )     qz   3 λz D    3 λz D   

at various values of λz is shown in the top right hand side diagram. It is this parameter that control the spanwise coherence (and thus, the length scale) of the vortices. The characteristic “lock-in” effect associated with vortex shedding induced dynamic response is controlled by the


154

aerodynamic damping parameter K a z . Establishing data of the mean wind velocity variation of

K a z will in general require wind tunnel experiments. As indicated in example 6.4 above, such data may often be fitted to an expression of the following type: −n −m     V  V    ⋅ exp  −   K a z = 2.6 ⋅ K az 0 ⋅    VR    VR   z    z   where V R z = ωz D ( 2π S t ) is the resonance velocity (see Eq. 5.32) and K a 0 is the value at the

apex of the K a z variation. See the lower left hand side diagram in Fig. 6.12, where n = 6 and m =8. Let us again consider a simply supported horizontal beam type of bridge with span L = 500m

that is elevated at a position z f = 50 m . Let us investigate the response variation with the mean wind velocity at various levels of structural eigen-damping. It is assumed that the entire span is flow exposed, i.e. L exp = L , and the expression for K a z given above is adopted. Let us allot the following values to the remaining constants that are necessary for a numerical calculation of σ rz ( x r = L 2 ) :

ρ (kg/m3)

B (m)

D (m)

1.25

20

4

mz

ωz

St

σˆ q z

bz

az

λz

K a0

(kg/m)

(rad/s) 0.8

0.1

0.9

0.15

0.4

1.2

0.2

10 4

Since m z is constant along the span, then the modally equivalent and evenly distributed mass m z = m z . The dynamic response is given in Eq. 6.91, i.e.:

σ rz ( x r = L 2 ) D

1

=

27 2 π 7 / 4

⋅

 ρ B D σˆ q z  m z

1/2

)

   

 1 − V /V 1 Rz ⋅ exp  −   2  bz 

  

⋅

2Dλ

⋅ S t 2  bz L ζ z − ζ ae z 

(

(

⋅ g z V Rz ,V

)

where

 V g z V R z ,V =   VR  z

(

)

3/2

   

The resonance mean wind velocity V R z is given by: V R z =

2

  

ωz D ≈ 5.1 m s . 2π S t

Under these circumstances the equation above may be rewritten into the following fourth order polynomial σˆ 4 − 1 − ζˆ σˆ 2 − βˆ 2 = 0 rz

(

)

rz

where

∫ φz d x 2

ζˆ =

4 m z

ρB

2

⋅

ζz K az

⋅

L

∫

L exp

φz2 d x

and

βˆ =

φz ( x r ) 2

5 2

π

7 4

  ρ D3 λ ⋅ ⋅ 2 b K  m z ∫ φz d x z a z L 

12

    

⋅

σˆ q z g z ⋅ S t 2 az

6.4 VORTEX SHEDDING

and where σˆ rz = σ rz

(az D ) .

155

Thus, the reduced standard deviation of the vortex shedding

induced dynamic response is given by 12

σˆ rz

12  2    1 − ζˆ  1 − ζˆ   2 ˆ = +   +β   2     2    

Fig. 6.16 Vortex shedding induced dynamic response The variation of σˆ rz with the mean wind velocity at three levels of structural eigen–damping is shown in Fig. 6.16. As can be seen, the vortex shedding induced dynamic response is self-limiting and strongly damping dependant. The maximum vortex shedding induced dynamic response will occur slightly above VR z , but for practical calculations the maximum value of σ rz may be obtained by setting V = V R z , in which case g z = 1 and K a z = K a 0 . As shown on the lower right hand side diagram in Fig. 6.15, the maximum value of σ rz is rapidly reduced with increased structural eigen-damping. If φ = [0 0 φθ ]

T

then 12

σˆ rθ

12  2    1 − ζˆ  1 − ζˆ   2 = +   + βˆ     2 2      


156

where σˆ rθ = σ rθ aθ and

∫ φθ d x 2

ζˆ =

4 m θ

ρB

4

⋅

ζθ K aθ

⋅

L

∫

L exp

2

φθ d x

and

βˆ =

φθ ( x r ) 2

5 2

π

7 4

  ρ D5 λ ⋅ ⋅ 2 b K  m θ ∫ φθ d x θ aθ L 

12

    

⋅

σˆ qθ gθ ⋅ S t 2 aθ

Chapter 7 DETERMINATION OF CROSS SECTIONAL FORCES

7.1 Introduction While we in chapter 6 focused exclusively on the determination of response displacements, we shall in this chapter deal with the determination of the corresponding cross sectional forces, i.e. the cross sectional stress resultants defined in chapter 1.3 (see Fig. 1.3.b). From a design point of view it is the maximum values of these quantities that decide the actual level of safety against structural failure. For a line like type of bridge structure the problem at hand is equivalent to that which is illustrated in Fig. 6.1, only that the response quantities we shall now set out to calculate are the cross sectional force components F (e.g. a bending moment, a torsion moment or a shear force) rather than the displacements which were in focus in chapter 6. The assumption of a Gaussian, stationary and homogeneous flow over the design period T (e.g. 10 min) is still valid, as well as the assumptions of linearity between load and load effects and a linear elastic structural behaviour. Thus, any cross sectional force component F may be described by the sum of its mean value and a fluctuating part that is Gaussian

Ftot ( x , t ) = F ( x ) + F ( x , t )

(7.1)

The time domain chain of events is illustrated in Fig. 7.1.a. Similar to that which was argued for the determination of displacements, it is in the following taken for granted that the fluctuating part of the cross sectional response forces are quantified by their standard deviation ( σ F ), as illustrated in Fig. 7.1.b. The maximum value of a force component at spanwise position x r is then given by Fm a x ( x r ) = F ( x r ) + k p ⋅ σ F ( x r )

(7.2)

where k p is the peak factor (that depends on the type of response process). The chain of events for cross sectional forces is equivalent to that which is shown for structural displacements in Fig. 6.2 because the assumption of linear elastic structural behaviour

160

7 DETERMINATION OF CROSS SECTIONAL FORCES

However, this does not imply that the fluctuations of cross sectional forces are insignificant. It only means that the resonant part of the force load effect is negligible (see chapter 2.10). For such a structure the total value of a cross sectional force component F at spanwise position x r may be obtained from

Fm a x ( x r ) ≈ F ( x r ) + k p ⋅ σ FB ( x r )

(7.4)

The entire solution, including σ FB , may then be obtained exclusively from static considerations, i.e. the determination of response spectra is obsolete. Since the solution contains the combined mean and fluctuating load effects, it represents the maximum value of the force load effect for a structure whose behaviour is defined as static. The more general solution, covering static as well as dynamic structural behaviour is given in Eq. 7.2. Having split the fluctuating part of the response into a background and a resonant part, the maximum value of F at x r may then be expressed by

Fm a x ( x r ) = F ( x r ) + k p ⋅ σ F2 B ( x r ) + σ F2 R ( x r )

(7.5)

where F and σ FB are obtained from static equilibrium conditions and σ FR is obtained from the resonant part of a modal frequency domain approach. For the determination of F the finite element type of approach that is shown below (chapter 7.2) is appropriate, unless the structural system is so simple that a direct analytical establishment of the equilibrium conditions is sufficient, in which case the solution is considered trivial. Similarly, for the determination of the background quasi-static part σ FB there are two alternatives. If the structural system is fairly complex a finite element approach is appropriate, but if the system is fairly simple a direct approach based on influence functions will suffice. Both methods are shown below (chapter 7.3). For the determination of the resonant part σ FR there is the possibility of establishing an equivalent load based on the inertia forces, i.e. the product of response acceleration and the oscillating mass variation, but this option is only useful if the structural system is very simple because the equivalent load pattern must reproduce the actual structural displacements that are relevant for the mode shapes that have been excited. In chapter 7.4 a more general procedure is given, based on the linear relationship between cross sectional stress resultants and the corresponding spanwise derivatives of the resonant displacement response. In a finite element formulation it is in the following assumed that the structural system has been modelled by nodes with six degrees of freedom as shown in Fig. 7.3 and by the use of beam or beam-column type of elements as shown in Fig. 7.4. At any level it is taken for granted that the load and load effect vectors can be split into a mean part and a fluctuating part, i.e. at a global system level

7.1 INTRODUCTION

 R1  R   2 R  Rtot (t ) =  3  = R + R (t ) R4  R   5  R 6 

 r1  r   2 r  rtot (t ) =  3  = r + r (t ) r4  r   5 r6 

and

161

(7.6)

and at the local level for an arbitrary element m

Ftotm

 F1  F   2 F  (t ) =  3  = Fm + Fm (t ) and  F4  F   5  F6 m

dtotm

 d1  d   2 d  (t ) =  3  = dm + dm (t ) d 4  d   5 d 6 m

(7.7)

The relationship between local forces and displacements is defined by the local stiffness matrix km , i.e.

Ftotm = km ⋅ dtotm

(7.8)

and the relationship between local and global degrees of freedom is defined by the matrix Am , i.e.

dtotm = Am ⋅ rtot

(7.9)

According to standard element method procedures the global stiffness matrix is then obtained by summation of contributions from all elements K = ∑ ATm ⋅ km ⋅ Am m

(7.10)

7.2 THE MEAN VALUE

7.2

163

The mean value

For the calculation of the mean value of cross sectional forces all quantities are time invariants and thus, Eq. 6.3 still holds, implying that the global displacements are given by r = K −1 ⋅ R

(7.11)

Similarly, the mean values of local forces and displacements (see Eqs. 7.8 and 7.9) are defined by Fm = km ⋅ dm  (7.12)  dm = Am ⋅ r 

and thus, the mean value of cross sectional forces is given by

(

)

Fm = km ⋅ ( Am ⋅ r ) = km ⋅  Am ⋅ K −1 ⋅ R   

(7.13)

Eqs. 7.8 – 7.13 are identical to that which one will usually encounter in an ordinary finite element formulation. The establishment of km and Am as well as the ensuing strategy for the calculation of global displacements and element force vectors may be found in many text books, see e.g. Hughes [25] or Cook et.al. [29]. Nonetheless, the brief summary presented above has been included for the sake of completeness. The only part that is special is the development of R , which has previously been shown in chapter 6.2.

7.3

The background quasi–static part

For the determination of the quasi-static part of the cross sectional response forces the mean part of the load as well as any motion induced contributions are obsolete. According to Eq. 5.8 the fluctuating part of the load on a line-like structure is given by q y ( x , t )    ρVB ˆ q ( x , t ) =  q z ( x , t )  = Bq ⋅ v = ⋅ Bq ⋅ v 2 q ( x , t )   θ 

(7.14)

where v and Bq are defined in Eqs. 5.9 and 5.12, and recalling that this was developed for a horizontal type of structure. As mentioned above the quasi-static part may be determined by a formal finite element formulation, or alternatively, by the use of static influence functions based on a direct establishment of the equilibrium conditions.

7.3 THE BACKGROUND QUASI-STATIC PART

165

where L exp is the flow exposed part of the structure and G M z is the static influence function for M z at x r (defined as the function containing the values of M z at x r when the system is subject to a unit load q y = 1 at arbitrary position x ). The variance of M z B is then defined by

2 σM z

B

2    2     ( x r ) = E  M z B ( x r ,t )  = E  ∫ G M z ( x ) ⋅ q y ( x ,t ) d x         L exp  

{

=

∫∫

L exp

}

(7.17)

G M z ( x1 ) ⋅ G M z ( x 2 ) ⋅ E q y ( x1 , t ) ⋅ q y ( x 2 , t )  d x1 d x 2

rendering a spatial and time domain averaging of the fluctuating cross sectional load. Introducing Eq. 7.15 then this space and time domain averaging is given by

E q y ( x1 , t ) ⋅ q y ( x 2 , t )  = 2

 D  ρVB  D  2  E  2 B C D u1 +  B C D′ − C L     where u1 = u ( x1 , t ) , u 2 = u ( x 2 , t )

   w 2    (7.18) and w1 = w ( x1 , t ) , w 2 = w ( x 2 , t ) . It is a usual    D D  w1  ⋅ 2 B C D u 2 +  B C D′ − C L    

assumption in wind engineering that cross-covariance between different velocity components is negligible, i.e. that E u ( x1 , t ) ⋅ w ( x 2 , t )  = E u ( x 2 , t ) ⋅ w ( x1 , t )  ≈ 0

(7.19)

in which case

E q y ( x1 , t ) ⋅ q y ( x 2 , t )  = 2 2 2   ρVB   D  D  ′   2 C E u x , t ⋅ u x , t + C − C ( ) ( )  1 2 L  E   2   B D   B D w ( x1 , t ) ⋅ w ( x 2 , t )          (7.20) Introducing (see chapters 2.2 and 3.3) E u ( x1 , t ) ⋅ u ( x 2 , t )  = σ u2 ⋅ ρu u ( ∆x )

where ρu u

(7.21) E w ( x1 , t ) ⋅ w ( x 2 , t )  = σ w2 ⋅ ρw w ( ∆x ) and ρw w are the covariance coefficients of the u- and w-components, and

where ∆x = x1 − x 2 is spanwise separation, then


166

E q y ( x1 , t ) ⋅ q y ( x 2 , t )  =  ρV 2 B   2

2 2 2    D  D    ′  ⋅  2 C D I u  ⋅ ρu u ( ∆x ) +  C D − C L  I w  ⋅ ρw w ( ∆x )      B   B 

where I u = σ u / V

and I w = σ w / V

(7.22)

are the u– and w–component turbulence

intensities. Thus, the variance of the background part is given by (see Eq. 7.17) 2

2 σM z

B

 ρV 2 B   ⋅ ∫∫ G M z ( x1 ) ⋅ G M z ( x 2 ) ⋅  2  L exp

( x r ) = 

2  D  D   2 C D I u  ⋅ ρu u ( ∆x ) +  C D′ − C L B   B 

2    I ρ ∆ x d x1 d x 2 ⋅ ( )  ww  w   

(7.23)

The volume integral in Eq. 7.23 represents a spatial averaging of the fluctuating load effect with respect to the bending component M z at a certain spanwise position x r . This is identical to that which has previously been dealt with in Chapter 2.10 (see Example 2.4). While Eq. 7.23 provides the calculation procedure for the background part of the cross sectional force component M z at x r alone, it is convenient to establish more general procedures comprising the background response of several components, e.g. the bending moments M y and M z as well as the torsion moment M x . These force components are in general given by G ( x ) ⋅ q ( x , t )  M x  θ  Mx    MB ( x r , t ) =  M y  = ∫ G M y ( x ) ⋅ q y ( x , t )  d x   L exp   G M ( x ) ⋅ q z ( x , t )   M z B z  

(7.24)

where G M n , n = x , y , z , are the static influence functions for cross sectional force components M x , M y and M z at x r . By adopting the definition

GM

 0  (x ) =  0  G M  z

it follows from Eqs. 7.14 and 7.24 that

0 GM y 0

GM x   0   0 

(7.25)


MB ( x r , t ) =

∫

GM ( x ) ⋅ q ( x , t ) d x =

L exp

ρVB 2

⋅

∫

167

{

}

ˆ ⋅ v ( x ,t ) d x GM ( x ) ⋅ B q

L exp

(7.26)

ˆ is the load coefficient matrix defined in Eq. 5.12, i.e. where B q 2 ( D / B ) C D  ˆ (x ) =  B 2C L q   2B CM 

( ( D / B ) C D′ − C L ) (C L′ + ( D / B ) C D )

(7.27)

 

′ B CM

T

and where v ( x , t ) = u ( x , t ) w ( x , t )  in the case of a horizontal bridge type of structure (see Eq. 5.9). The background covariance matrix Cov M x M z   Cov M y M z   2 σM zMz  B

(7.28)

T  2          ρVB  ˆ ⋅ v dx  ⋅  ˆ ⋅ v dx   GM ⋅ B = E   ∫ GM ⋅ B q q  ∫   2    L exp     L exp  

(7.29)

Cov M M B

 σ2  MxMx ( x r ) = Cov M y M x  Cov M z M x

Cov M x M y 2 σM yM y

Cov M z M y

is then obtained from Cov M M B ( x r ) = E MB ( x r , t ) ⋅ MTB ( x r , t ) 

(

 ρVB  =   2 

2

∫∫

L exp

)

(

{

)

}

T ˆ ⋅ E  v ( x , t ) ⋅ vT ( x , t )  ⋅ B ˆT G M ( x1 ) ⋅ B 1 2 q   q ⋅ G M ( x 2 ) d x1 d x 2

Introducing Eq. 7.21 and adopting the assumptions in Eq. 7.19, then σ u2 ⋅ ρu u ( ∆x )  0 E  v ( x1 , t ) ⋅ vT ( x 2 , t )  ≈   = V 2 ⋅ Iv2 ⋅ ρ ( ∆x ) 0 σ w2 ⋅ ρw w ( ∆x )  

{

}

(7.30)

where I Iv =  u 0

0  I w 

and

ρ ρv ( ∆ x ) =  u u  0

0  ρw w 

(7.31)


168

and thus,

Cov M M B

 ρV 2 B ( x r ) =   2

  

2

∫∫

L exp

{

}

T ˆ ⋅ I2 ⋅ ρ ( ∆x )  ⋅ B ˆT G M ( x1 ) ⋅ B q v v  q ⋅ G M ( x 2 ) d x1 d x 2

(7.32) The covariance matrix in Eq. 7.32 will be symmetric because x1 and x 2 are interchangeable and ρu u and ρw w are only functions of the separation ∆x = x1 − x 2 . In a fully expanded format the variance of the background response components are given by

σ 2   MxMx  2 σ 2  =  ρV B M M   y y  2  2  σ M z M z  B

  

2

g ( x , x )  MxMx 1 2  ∫∫  g M y M y ( x1 , x 2 ) d x1 d x 2 L exp  g M M ( x1 , x 2 )   z z 

(7.33)

where

(

g M x M x = B 2 G M x ( x 1 ) G M x ( x 2 )  2C M I u 

 g M y M y = G M y ( x 1 ) G M y ( x 2 )  2C L I u 

(

)

2

)

2

′ I w ) ρw w ( ∆x )  (7.34) ρu u ( ∆ x ) + ( C M  2





ρu u ( ∆x ) +  C L′ + 

2  D   C D  I w  ρw w ( ∆x )  B    (7.35)

g M z M z = G M z ( x1 ) G M z ( x 2 ) ⋅ 2  D  D   2 C D I u  ρu u ( ∆x ) +  C D′ − C L B   B 

2    ρ ∆ I x ( )   w  ww   

(7.36)

Similarly, the corresponding covariance between background components may be expanded into Cov M M  x y    ρV 2 B Cov M M  =  x z  2    Cov M y M z   B where

  

2

 g M M ( x1 , x 2 )   x y   g x x , ( ) ∫∫  M x M z 1 2  d x1 d x 2 L exp  g M y M z ( x1 , x 2 )   

(7.37)


g M x M y = B G M x ( x1 ) G M y ( x 2 ) 4 C L C M I u2 ρu u ( ∆x )  D   ′ I w2 ρw w ( ∆x )  +  C L′ + C D  C M B   

169

(7.38)

 D g M x M z = B G M x ( x1 ) G M z ( x 2 ) 4 C D C M I u2 ρu u ( ∆x )  B  D  ′ I w2 ρw w ( ∆x )  +  C D′ − C L  C M B   

(7.39)

 D g M y M z = G M y ( x1 ) G M z ( x 2 ) 4 C D C L I u2 ρu u ( ∆x )  B  D D   +  C D′ − C L  C L′ + C D  I w2 ρw w ( ∆x )  B B   

(7.40)

Example 7.1: Let us set out to calculate the variances and covariance of the torsion and bending moments M x , M y and M z at midspan of the simply supported beam type of bridge illustrated in Fig. 7.6. Let

us for simplicity assume that it has a typical bridge type of cross section where C D′ , C L and C M are negligible and C D ⋅ D B σ 2   MxMx  2 σ 2  =  ρV B M M  y y  2  2   σ M z M z 

C L′ . Then

   BC′ I 2 ⋅G  ⋅ ⋅ ∆ x G x ρ x ( ) ( ) ( ) ( ) M w Mx Mx ww 1 2  2 LL    2  ⋅ ∫ ∫  (C L′ I w ) ⋅ G M y ( x1 ) ⋅ G M y ( x 2 ) ⋅ ρw w ( ∆x ) d x1d x 2  00  2  D   2 C I G x G x ρ x ⋅ ⋅ ⋅ ∆ )  D u Mz ( 1) Mz ( 2) uu ( B    

Cov M M  x y 2   Cov M M  =  ρV B x z   2   Cov M y M z   

 B C L′ C M ′ I w2 ⋅ G M ( x1 ) ⋅ G M ( x 2 ) ⋅ ρw w ( ∆x )  2 x y   LL d x1d x 2 0  ⋅ ∫ ∫    00 0    

where it has been taken for granted that L exp = L .


Jˆ 2   MxMx  Jˆ 2 = 1  M yM y  N 2  ˆ2   J M z M z 

171

 ρw w ( xˆ n − xˆ m  ∑ ∑ ψ ( xˆn ) ⋅ψ ( xˆm ) ⋅  ρw w ( xˆn − xˆm  n =1 m =1  ρu u ( xˆ n − xˆ m N

N

) )  ) 

and it is just a matter of choosing N sufficiently large. Let us assume that the position of the bridge is at z f = 50 m and that the relevant length scales of the u and w components are given by (see Eq 3.36): xf

(

L u = 100 z f /10

)

0.3

yf

= 162 m ,

Lu =

xf

L u / 3 = 54 m

and

yf

Lw =

xf

L u /16 = 10 m ,

such that

ρu u ( ∆xˆ ) = exp ( −cu ⋅ ∆xˆ )

where

cu = L

yf

Lu

ρw w ( ∆xˆ ) = exp ( −cw ⋅ ∆xˆ )

where

cw = L

yf

Lw

Let us for simplicity set L =

xf

L u and N = 5 (which in general will be far too small, as shown in

Fig. 7.7). Thus, cu = 3 , cw = 16 . The position vector and the influence function are given by xˆ = [0.1 0.3 0.5 0.7 0.9 ]

T

ψ ( xˆ ) = [0.1 0.3 0.5 0.3 0.1 ]

T

The influence function multiplications ψ ( xˆ n ) ⋅ψ ( xˆ m ) are then given by

ψ ( xˆ n )

ψ ( xˆ m )

0.1 0.3 0.5 0.3 0.1

0.1

0.3

0.5

0.3

0.1

0.01 0.03 0.05 0.03 0.01

0.03 0.09 0.15 0.09 0.03

0.05 0.15 0.25 0.15 0.05

0.03 0.09 0.15 0.09 0.03

0.01 0.03 0.05 0.03 0.01

while the covariance coefficients associated with the u and w components are given by:

ρu u ( ∆xˆ ) :

xˆ n 0.1

xˆ m

0.1 0.3 0.5 0.7 0.9

1 0.549 0.301 0.165 0.091

0.3

0.5

0.7

0.9

0.549 1 0.549 0.301 0.165

0.301 0.549 1 0.549 0.301

0.165 0.301 0.549 1 0.549

0.091 0.165 0.301 0.549 1

xˆ n


172 ρw w ( ∆xˆ ) :

0.1 1 0.0408 0.0017 0.0001 ≈0

0.1 0.3 0.5 0.7 0.9

xˆ m

xˆ n 0.5 0.0017 0.0408 1 0.0408 0.0017

0.3 0.0408 1 0.0408 0.0017 0.0001

0.7 0.0001 0.0017 0.0408 1 0.0408

0.9 ≈0 0.0001 0.0017 0.0408 1

The inner products ψ ( xˆ n ) ⋅ψ ( xˆ m ) ⋅ ρu u ( ∆xˆ ) and ψ ( xˆ n ) ⋅ψ ( xˆ m ) ⋅ ρu u ( ∆xˆ ) are then: n

1

2

4

5

10 2 ⋅ψ ( xˆ n ) ⋅ψ ( xˆ m ) ⋅ ρu u ( ∆xˆ )

m

1 2 3 4 5 n

1 1.647 1.505 0.495 0.091

1.647 9 8.235 2.709 0.495

1.505 8.235 25 8.235 1.505

0.495 2.709 8.235 9 1.647

0.091 0.495 1.505 1.647 1

1

2

3

4

5

0.0003 0.0153 0.612 9 0.1224

≈0 0.0003 0.0085 0.1224 1

10 2 ⋅ψ ( xˆ n ) ⋅ψ ( xˆ m ) ⋅ ρw w ( ∆xˆ )

m 1 2 3 4 5

3

1 0.1225 0.0085 0.0003 ≈0

0.1225 9 0.612 0.0153 0.0003

0.0085 0.612 25 0.612 0.0085

The normalised joint acceptance functions are given by 1

1 5 5 2 Jˆ M z M z =  2 ⋅ ∑ ∑ ψ ( xˆ n ) ⋅ψ ( xˆ m ) ⋅ ρu u ( ∆xˆ )  ≈ 0.2  5 n =1 m =1  1

Jˆ M x M x = Jˆ M y M y

1 5 5 2 =  2 ⋅ ∑ ∑ ψ ( xˆ n ) ⋅ψ ( xˆ m ) ⋅ ρw w ( ∆xˆ )  ≈ 0.14  5 n =1 m =1 

and thus 1 2



′ Iw  σ M x M x ≈ 0.14 ⋅  ρV 2 B 2 L C M

σM yM y



1  ≈ 0.07 ⋅  ρV 2 B L2C L′ I w  2  1 2



σ M z M z ≈ 0.2 ⋅  ρV 2 DL2C D I u  This solution has been based on L exp = L =



yf

L u and N = 5 . If N = 40 then the integration

coefficient 0.14 is reduced to 0.11, the coefficient 0.07 is reduced to 0.055 while the coefficient


173

0.2 is reduced to 0.188. The problem of choosing a sufficiently large number of integration points is illustrated in Fig. 7.7, where the normalised joint acceptance function Jˆ for an arbitrary force nn

component whose influence function is linear and with a maximum of 0.5 at midspan is plotted versus N for three different values of the ratio between L exp and the relevant length scale x L j , j = u or w (and x = y f ). It is seen that the necessary number of integration points is in general

considerable. The reason for this is that ρ jj ( j = u or w ) is a rapidly decaying function. Similarly, in Fig. 7.8 the joint acceptance function has been plotted versus the ratio between the length of the span ( L exp = L ) and the relevant integral length scale x L j , j = u or w . The case

(

)

lim L / x L j → 0 is identical to the situation with an evenly distributed load along the entire span. As can be seen, Jˆ n n is a rapidly decreasing function with increasing values of L large value of L

Fig. 7.7

x

x

L j . At a

L j it is close to 0.05.

The joint acceptance function Jˆ n n vs. number of integration points


174

Fig. 7.8

The joint acceptance function Jˆ n n at various span length ( L exp = L )

While the solution strategy based on influence functions shown above is suitable for many cases of fairly simple structural systems, a formulation within the finite element method is more suitable for a general approach. Recalling that Eq. 5.8 was developed for a horizontal type of structure, it is seen from Fig. 7.9 that in general v = [u w ]

T

for a

horizontal element and v = [u v ] for a vertical element. Defining the non-dimensional T

instantaneous fluctuating wind velocity vector at an arbitrary node p

vˆ p =

 u (t )  1   v (t )  V  w (t )   

(7.41)

and the matrix

 1 0 0     for a h or ison t a l elem en t  0 0 1  ψm =  (7.42)  1 0 0  for a ver t ica l elem en t  0 1 0     which is associated with the direction of an adjoining element m , it is seen from Fig. 7.9 that the distributed load vector acting on this element is given by


176

The contribution from element m to concentrated loads in node p is then given by (see Fig. 7.9.a) Q y   ρV 2   B L  L   ˆ ⋅ ψ ⋅ vˆ Q pm (t ) = Q z  = qm ( x , t ) ⋅ m =  ⋅ ⋅B q m p (7.44)  2   2  2 m  p Q   θ

(

)

pm

where L m is the element length, and where it has been assumed that the nodal discretisation is such that all wind velocity properties with sufficient accuracy may be allotted to node p , and that they are constants along the span of the element. (This is a simplification that is not mandatory, but otherwise, an element integration scheme has to be adopted.) Comparing the definition of nodal loads shown in Fig. 7.3 to the element load components shown in Fig. 7.9, it is seen that the contribution from element m to the load vector in node p is defined by (see Fig. 7.9.b and c)

R pm

 R1  R   2 R3  =  R4  R   5  R 6  p

   =  

0 Q y

Qz

−Qθ

T

0 0  for a h or izon t a l elem en t (7.45)

 −Q z

Qy

0 0 0 −Qθ 

T

for a ver t ica l elem en t

m

and thus,  ρV 2 R pm (t ) = θm ⋅ Q pm =    2

  BL  ˆ  ⋅   θm ⋅ Bqm ⋅ ψm ⋅ vˆ p   p  2 m

(

)

(7.46)

where indices m and p indicate quantities associated with element m at node p , and where

θm

     =     

0 1 0 0 0 0  0 0 1 0 0 0    0 0 0 −1 0 0 

T

0 1 0 0 0 0   −1 0 0 0 0 0     0 0 0 0 0 −1 

for a h or izon t a l elem en t (7.47)

T

for a ver t ica l elem en t


177

Thus, the load vector in node p is obtained by adding up the contributions from all adjoining elements. i.e.  ρV 2 R p (t ) = ∑ R pm =   m  2

   BL  ˆ   ∑   θ ⋅ Bq ⋅ ψ 2   m p  m

(

 ρV 2 ˆ   ⋅ Q ⋅ vˆ  vˆ p =     m   2 p

)

(7.48)

where

ˆ =  BL ⋅ θ ⋅B ˆ ⋅ψ Q ∑  2 p q  m m

(7.49)

The total system load vector is then given by R (t ) = R1

Rp

R N 

T

(7.50)

where N is the total number of nodes. Since the content of this load vector is considered quasi-static the relationship K ⋅ r (t ) = R (t ) holds, and because u (t ) and w (t ) are both zero mean variables then R (t ) as well as r (t ) are also zero mean variables. Thus, it is seen from to Eqs. 7.6 – 7.9 that the fluctuating background quasi-static part of the element force vector Fm (t ) is given by  F1  F   2 F  Fm (t ) =  3  = km ⋅ dm (t ) = km ⋅  Am ⋅ r (t )  = km ⋅ Am ⋅ K −1 ⋅ R (t )   F4  F   5  F6 m

{

}

(7.51)

The covariance matrix between cross sectional force components

Cov Fm Fm

σ F2  1    =      

Cov F1 F2

Cov F1 F3

Cov F1 F4

Cov F1 F5

σ F22

Cov F2 F3

Cov F2 F4

Cov F2 F5

σ F23

Cov F3 F4

Cov F3 F5

σ F24

Cov F4 F5

S ym .

σ F25

Cov F1 F6   Cov F2 F6   Cov F3 F6   (7.52) Cov F4 F6   Cov F5 F6  σ F26 


178 is then defined by

{

 Cov Fm Fm = E Fm ⋅ FmT  = E  km  Am     = km ⋅  Am ⋅  K −1 ⋅ E R ⋅ RT    = k m ⋅  Am 

(K R )} ⋅ {k −1

( )

−1  ⋅ K

( )

 ⋅ K −1 ⋅ Cov R R ⋅ K −1 

T

T

m

(

)}

 A K −1 R   m 

T

 T  T  ⋅ Am  ⋅ k m  

   (7.53)

 T  T  ⋅ Am  ⋅ km

where the 6N by 6N nodal load covariance matrix

Cov R R

  = E R ⋅ RT  =   

Cov R p R k

   where  

p  = 1,2,3,… , N k

(7.54)

Its content is N nubers of 6 by 6 covariance matrices between force components associated with nodes p and k , each is given by

Cov R p R k

= E R p ⋅ RTk  = E

 2 ˆ ⋅ vˆ   ρV ⋅ Q    2 p 

 ρV 2 =  2 

  p

 ρV 2 ⋅  2 

 ˆ T ˆT  ⋅ Q p ⋅ E  vˆ p ⋅ vˆ k  ⋅ Q k k

 ρV 2 =  2 

  p

 ρV 2 ⋅  2 

 ˆ ˆ ov ˆT  ⋅ Q p ⋅ C v p vk ⋅ Q k k

T

 ρV 2 ˆ  ⋅ ⋅ Q ⋅ vˆ     2 k

    (7.55)

where

ˆ Cov v p vk = E

 vˆ p ⋅ vˆ Tk 

1

=  V2

u puk  ⋅ E  v puk  w p u k

u pvk v p vk w pvk

u pw k   v pw k   w p w k 

(7.56)

As previously mentioned (see Eq. 7.19), it is a usual assumption in wind engineering that cross-covariance between different velocity components is insignificant, i.e. that all off diagonal terms in Eq. 7.56 may be neglected, in which case ˆ Cov v p v k ≈ I p ⋅ Ik ⋅ ρ pk

(7.57)

where I j = d iag [ I u

Iv

I w ]j

p j= k

(7.58)


182

ˆ Cov R2 R4

0 0 0 0 0 0.002164 0 0  0 0 0.035 −0.21 = 0 −0.21 1.26 0 0 0 0 0  0 0 0 0 

0 0 0 0 0 0

0 0  0  0 0  0 

The covariance matrix of a chosen set of cross sectional forces is given in Eq. 7.53.

7.4

The resonant part

Fluctuating cross sectional forces at an arbitrary spanwise position x r that are exclusively ascribed to the resonant part of the response may in general be extracted from the derivatives of predetermined modal displacements

r ( x r , t ) = Φr ( x r ) ⋅ η (t )

(7.62)

as defined in Eqs 4.7 and 4.8 (see also Eq. 4.79). The direct transition from the variance of the fluctuating displacement response quantities to the variances of corresponding dynamic cross sectional forces is therefore presented below. The procedures for the calculation of response displacements are in a general format shown in Chapter 4. For the special cases of buffeting or vortex shedding induced dynamic response the procedure is shown in Chapter 6. For simplicity it is in the following as usual assumed that we are dealing with a line-like horizontal (bridge) type of structure where axial forces may be disregarded, in which case the force component F1 in Eq. 7.7 may be omitted. (Axial forces may in general be determined by the product of the axial stiffness of a beam type of element and the difference between the axial displacements at its end nodes.) It follows from the definition of cross sectional forces in Fig. 1.3 that the connection between the fluctuating force vector at an arbitrary spanwise position x r and the corresponding cross sectional stress resultant is F ( x r , t ) = [ F2

F3

F4

F5

F6 ] = V y T

Vz

Mx

My

M z 

T

(7.63)

where V y ,V z are the shear forces in the direction of the y and z axes, M y , M z are the

bending

moments

about

the

same

axes,

and

where

Mx

is

7.4 THE RESONANT PART

183

the torsion moment. It is taken for granted that the material behaviour is linear elastic and that displacements are small (i.e. there is no geometric non-linearity). The relationship between cross sectional stress resultants and the derivatives of the corresponding displacements are then given by the following differential equations (see e.g. Chen & Atsuta [27]):

M x ( x r , t ) = GI t ⋅ rθ′ ( x r , t ) − E I w ⋅ rθ′′′( x r , t )   M y ( x r , t ) = −E I y ⋅ rz′′ ( x r , t )   M z ( x r , t ) = E I z ⋅ ry′′ ( x r , t )   V y ( x r , t ) = − M z′ ( x r , t ) = − E I z ⋅ ry′′′( x r , t )   V z ( x r , t ) = M y′ ( x r , t ) = − E I y ⋅ rz′′′( x r , t ) 

(7.64)

where the prime behind symbols indicate derivation with respect to x . Defining the cross sectional property matrix T ( x r )

 0  0  T (xr ) =  0   0  E I z

−E I z 0

0 0

0 −E I y

0 0

0 0

0 −E I y

0 0

GI t 0

0

0

0

0

   −E I w   0   0  0 0

(7.65)

then F ( x r , t ) as defined in Eq. 7.63 is given by F ( x r , t ) = T ⋅ ry′′ ry′′′ rz′′ rz′′′ rθ′

rθ′′′

T

(7.66)

Multi mode approach

Introducing the six by N m od mode shape derivative matrix

φ y′′1  φ y′′′1  φz′′ 1 β(xr ) =  φz′′′  1 φθ′  1 φθ′′′  1

φ y′′i

φ y′′N

m od

φ y′′′i

φ y′′′N

m od

φz′′i

φz′′N

m od

φz′′′i

φz′′′N

m od

φθ′i

φθ′N

m od

φθ′′′

φθ′′′

i

N m od

           

(7.67)


184

where i is an arbitrary mode number and N m od is the total number of modes, it then follows from Eq. 7.62 that

F ( x r , t ) = T ( x r ) ⋅ β ( x r ) ⋅ η (t ) 

(7.68)

Taking the Fourier transform on either side

a F ( x r , ω ) = aV y 

aV z

where aη (ω ) =  aη1  S V y V y    SF ( x r ,ω ) =     

aM x

aηi

aM y

aηN

T

a M z  = T ( x r ) ⋅ β ( x r ) ⋅ aη (ω )  

(7.69)

T

 , and defining the matrix 

m od

S V yV z

SVyM x

SVyM y

S VzVz

S Vz M x

S Vz M y

S MxMx

S MxMy

S ym .

SVyM z   S Vz M z   S MxMz   S M yM z   S M z M z 

S M yM y

(7.70)

containing auto spectral densities and cross spectral densities of all force components, then the following is obtained: 1 * T 1 a F ⋅ a F = lim T →∞ π T T →∞ π T 1  ⇒ S F ( x r , ω ) = T ⋅ β ⋅  lim aη* ⋅ aηT →∞ T π T 

S F ( x r , ω ) = lim

(

(

)

(

)

 T ⋅ β ⋅ a*  ⋅ T ⋅ β ⋅ a  η   η  

) ⋅ β

T

T

⋅ TT = T ⋅ β ⋅ Sη ⋅ βT ⋅ TT

(7.71)

where Sη is given in Eq. 4.74. However, because Sη contains the entire dynamic response, i.e. background as well as resonant, it requires reduction to include only the resonant part. The extraction of the resonant part is equivalent to a white noise type of load assumption, and thus ˆ * (ω ) ⋅ S ⋅ H ˆ T (ω ) SηR (ω ) = H η η Qˆ R

where Hˆ η is given in Eqs. 4.69 and where (see Eq. 4.75)

(7.72)


185

     SQˆ =  S Qˆ Qˆ R i j     whose elements on row i column j are given by

∫∫

φTi ( x1 ) ⋅ Sqq ( ∆x , ωi ) ⋅ φ j ( x 2 ) d x1 d x 2

L exp

S Qˆ Qˆ =

(7.73)

(7.74)

(ω M ) ⋅ (ω M ) 2 i

i j

i

2 j

j

where ∆x = x1 − x 2 , and where Sqq ( ∆x , ωi ) is the spectral density matrix of cross

sectional loads at the eigen-frequency ωi (see Eq. 4.78), i.e. S  q yqy Sqq ( ∆x , ωi ) =  S q z q y  S q q  θ y It is seen that SQˆ

S q y qz S qz qz S qθ q z

S q y qθ   S q z qθ   S qθ qθ  

(7.75)

is frequency independent. The resonant part of the auto and cross

R

spectral density matrix of all force components is then given by ˆ * (ω ) ⋅ S ⋅ H ˆ T (ω )  ⋅ βT ⋅ TT (7.76) S FR ( x r , ω ) = T ⋅ β ⋅ SηR (ω ) ⋅ βT ⋅ TT = T ⋅ β ⋅ H η Qˆ R   η

The corresponding matrix containing the resonant part of the variance and covariance of cross sectional stress resultants is obtained by frequency domain integration, i.e. ∞

Cov FFR ( x r ) = ∫ S FR ( x r , ω ) d ω 0

σ V2 V  y y    =    

CovV yV z

CovV y M x

CovV y M y

σ V2z V z

CovV z M x

CovV z M y

2 σM xMx

Cov M x M y

S ym .

2 σM yM y

CovV y M z   CovV z M z    Cov M x M z   Cov M y M z   2 σM  M z z  R

(7.77)


186

If the load case is wind buffeting as described in chapter 6.3, and the assumption of negligible cross spectra between fluctuating flow components is adopted, then ˆ ( ∆x , ω ) , see Eq. 6.40. Thus, Sqq ( ∆x , ω ) is given in Eq. 6.57 and Sv ( ∆x , ω ) = V 2Iv S v

 ρV 2 B Sqq ( ∆x , ωi ) =   2 

2

 ˆ ⋅ I2 ⋅ Sˆ ( ∆x , ω )  ⋅ B ˆT  ⋅ B q v v i  q 

(7.78)

where Iv = d iag [ I u

I w ] and

ˆ ˆ ( ∆x , ω ) = S u u S v i  0

 S u (ωi ) ˆ  ⋅ Cu u ( ∆x , ωi ) 0   2 0   σu  =  ˆ S w (ωi ) ˆ S w w    C x ω ⋅ ∆ 0 , ww ( i)   σ w2  

(7.79)

and where Cˆ u u ( ∆x , ωi ) and Cˆ w w ( ∆x , ωi ) are the reduced co-spectra defined in Eq.6.64. Introducing

the

(

)

evenly

distributed

and

(

)

modally

equivalent

masses

m i = M i / ∫ φTi ⋅ φi d x and m j = M j / ∫ φTj ⋅ φ j d x (see Eq. 6.41), then the content of L

SQˆ

R

L

on row i column j is given by 2

S Qˆ Qˆ

i j

ρB 3 ρB 3  V   V ⋅ ⋅ (ωi ) =  ⋅ 2m i 2m j  B ωi   B ω j

where Jˆ ij2

(ωi ) =

∫∫

L exp

{

2

 2  ⋅ Jˆ ij (ωi )  

(7.80)

}

ˆ ( ∆x , ω )  ⋅ B ˆ ⋅ I2 ⋅ S ˆ T ⋅ φ (x )d x d x φTi ( x1 ) ⋅ B q v v i  q j 2 1 2  T     ∫ φi ⋅ φi d x  ⋅  ∫ φTj ⋅ φ j d x      L  L 

(7.81)

Single mode three component approach

In many cases where eigen-frequencies are well separated and flow induced coupling effects are negligible a multi mode procedure as presented above may with sufficient accuracy be replaced by a mode by mode approach. Then all modes are uncoupled, and therefore, the covariance contributions between force components from different modes


187

are zero, and thus, the total covariance of cross sectional forces may be obtained as the sum of the covariance contributions from each mode, i.e. Cov FFR ( x r ) =

N m od

∑ i =1

Cov FFR

(7.82)

i

For an arbitrary mode i Eq. 7.76 is then reduced to

S FR

i

( x r , ω ) = T ⋅ βi ⋅  Hˆ i (ω )

2



where βi = φ y′′ φ y′′′ φz′′ φz′′′ φθ′

 ⋅ S Qˆ Qˆ (ωi )  ⋅ βTi ⋅ TT i i 

T

φθ′′′ , and thus

∞

Cov FFR

i

(7.83)

( x r ) = ∫ S FRi ( x r , ω ) d ω = T ⋅ βi ⋅ βTi 0

∞

⋅ TT ⋅ S Qˆ Qˆ ⋅ ∫ Hˆ i (ω ) d ω i i

0

(7.84) =

(

πωi ⋅ S Qˆ Qˆ

)(

i i

4 1 − κ aei ⋅ ζ i − ζ aei

)

⋅ T ⋅ βi ⋅ βTi ⋅ TT

where κ aei and ζ aei are given in Eqs. 6.45 and 6.46, and where

S Qˆ Qˆ

i i

∫∫

Jˆ ii2 =

L exp

 ρB 3 =  2m i 

2  V  ˆ  ⋅  ⋅ J ii    B ωi  

{

2

(7.85)

}

ˆ ( ∆x , ω )  ⋅ B ˆ ⋅ I2 ⋅ S ˆ T ⋅ φ (x )dx dx φTi ( x1 ) ⋅ B q v v i  q i 2 1 2  T   ∫ φi ⋅ φi d x    L 

2

Performing the multiplication T ⋅ βi ⋅ βTi ⋅ TT and then the following is obtained:

(7.86)


188 T ⋅ βi ⋅ βTi ⋅ TT =   φ ′′′E I  y z              

(

) (φy′′′E I z )(φz′′′E I y ) 2

(φz′′′E I y )

{(

)

− φ y′′′E I z ⋅

(φθ′ GI t − φθ′′′E I w )}

{(

)

− φz′′′E I y ⋅

2

(φθ′ GI t − φθ′′′E I w )} (φθ′ GI t − φθ′′′E I w )2

S ym .

(φy′′′E I z )(φz′′E I y ) (φz′′′E I y )(φz′′E I y ) {(φθ′ GI t − φθ′′′E I w )

( )} 2 (φz′′E I y )

⋅ φz′′E I y

(

− φ y′′′E I z



)(φ y′′E I z )

  − φz′′′E I y φ y′′E I z   ′′′ ′ GI φ E I φ − {( θ t θ w )   ⋅ φ y′′E I z   − φz′′E I y φ y′′E I z   2  φ y′′E I z  i

(

)(

)}

(

(

)

)(

(

)

)

(7.87) It is readily seen from Eq. 7.87 that the covariance matrix associated with an arbitrary mode i has the properties

Cov FFR σ V2 V  y y        

i

(xr ) = σ V yV y ⋅ σ V z V z

−σ V yV y ⋅ σ M x M x

σ V yV y ⋅ σ M y M y

σ V2z V z

−σ V z V z ⋅ σ M x M x

σVzVz ⋅ σ M y M y

2 σM xMx

σ M x M x ⋅σ M yM y 2 σM yM y

S ym .

−σ V yV y ⋅ σ M z M z   −σ V z V z ⋅ σ M z M z    σ M x M x ⋅σ M z M z   −σ M y M y ⋅ σ M z M z   2 σM  zMz i (7.88)

I.e., for an arbitrary mode i

ρm n i =

Covm ni

σ m m i ⋅ σ n ni

 +1 for m n = V y V z , V y M y , V z M y , M x M y , M x M z = (7.89) −1 for m n = V y M x , V y M z , V z M x , V z M z , M y M z

which could be expected, because within a single mode all coupling between cross sectional force components are caused by the structural properties already contained in the relevant mode shape, and thus, all covariance coefficients will either be plus or minus unity (depending on the chosen sign conventions). Thus, the problem is reduced to the calculation of variance contributions from each of the modes that have be


189

considered necessary for a sufficiently accurate solution. It follows from Eqs. 7.84 – 7.87 that

 σ V yV y     σVzVz    ρB 3 σ M x M x  = 4m i   σ M y M y    σ M z M z  i

 V  ˆ  πωi   J ii   1 −κ  B ωi  aei ⋅ ζ i − ζ aei 

1

2

(

)(

)

2   

 φ y′′′E I z   φz′′′E I y   φθ′ GI t − φθ′′′E I w  φz′′E I y   φ y′′E I z 

         i

(7.90)

where Jˆ ii is given in Eq. 7.86. The total variances and covariance coefficients are then given by Eqs. 7.81 and 7.88.

Single mode single component approach

In some cases a single mode single component approach will suffice. The necessary calculations are then further reduced. Let us first consider a single mode that only T

contains an along wind y component, i.e. φ = φ y 0 0  , and whose eigen-frequency is ω y . Then the necessary calculations are reduced to

 σ V yV y  σ M z M z

 ρB 3 =  4 m y

 V ⋅  B ωy 

2

  ⋅ Jˆ y ω y  

( )

 πω y  ⋅ 1 − κ ae y ⋅ ζ y − ζ ae y 

(

)(

1

)

2   

 φ y′′′E I z ⋅  φ y′′E I z 

  (7.91)  

where m y is defined in Eq. 6.20, κ ae y and ζ ae y are defined in Eq. 6.24, and where Jˆ y is given in Eq. 6.22 (see also Eq. 6.19). Similarly, if the relevant mode only contains an across wind z component, i.e. φ = [ 0 φz

0 ] , whose eigen-frequency is ωz , then T

  σ V z V z  ρ B 3  V 2 πωz  = ⋅  ⋅ Jˆ z (ωz ) ⋅  σ  4 ω m B 1 − κ aez ⋅ ζ z − ζ aez z  z   M y M y  

(

)(

1

)

2   

 φz′′′E I y ⋅  φz′′E I y 

  (7.92)  

where m z is defined in Eq. 6.27, κ aez and ζ aez are defined in Eq. 6.31, and where Jˆ z is given in Eq. 6.28.


190

Finally, if the relevant mode only contains a cross sectional rotation component θ , i.e. φ = [0 0 φθ ] , whose eigen-frequency is ωθ , then T

σMxMx

2  πωθ ρB  V  ˆ = ω J ( )   θ θ   1 −κ 4 m θ  B ωθ  aeθ ⋅ ζ θ − ζ aeθ 

1

4

(

)(

)

2  φθ′ GI t − φθ′′′E I w (7.93)  

where m θ is defined in Eq. 6.27, κ aeθ and ζ aeθ are defined in Eq. 6.31, and where Jˆθ is given in Eq. 6.29.

Example 7.3: Let us again consider the simply supported beam shown in Fig. 7.6, and as usual, let us for simplicity assume that all cross sectional quantities are constants along the span of the bridge, and that it has a typical bridge type of cross section where C D′ , C L and C M are negligible and C L′ . Let us set out to determine the covariance matrix associated with cross sectional

CD ⋅ D B

forces at spanwise positions x r = 0 and x r = L / 2 that is caused by resonant oscillations in a chosen mode

φ y  a y      π φi = φz  =  a z  ⋅ sin x L φ     θ  i  aθ  whose eigen-frequency, eigen-damping-ratio and modally equivalent and evenly distributed mass are ωi , ζ i and m i . The necessary calculations are given in Eq. 7.83, i.e.

Cov FFR

i

(xr ) =

(

πωi ⋅ S Qˆ Qˆ

)(

i i


)

⋅ T ⋅ βi ⋅ βTi ⋅ TT

where S Qˆ Qˆ and T ⋅ βi ⋅ βTi ⋅ TT are given in Eqs. 7.84 and 7.86. Since (see Ex. 7.2) i i

 D 2 B C D ˆ =  0 B q  0  

 0   C L′  ′  B CM  

and

 2ˆ ˆ ( ∆x , ω ) =  I u S u u Iv2 ⋅ S v i  0

0 I w2 Sˆw w

  


191

then J ii2 =

2   D  2 C I  ∫∫  B D u  ⋅ φyi ( x1 ) ⋅ φyi ( x 2 ) ⋅ Sû u ( ∆x , ωi ) + L exp  

(C ′ I )2 ⋅ φ ( x ) ⋅ φ ( x ) + ( B C ′ I )2 ⋅ φ ( x ) ⋅ φ ( x )  ⋅ Sˆ ( ∆x , ω ) + θi θi 1 2 1 2  zi zi M w i  L w  ww ′ I w2 φz ( x1 ) ⋅ φθ ( x 2 ) + φθ ( x1 ) ⋅ φz ( x 2 )  ⋅ Sˆw w ( ∆x , ωi ) d x1d x 2 C L′ B C M i i i  i 

}

and Jˆ ii =

∫(

L

J ii

φ y2i

)

+ φz2i + φθ2i d x

Introducing the sinusoidal mode shapes, and S (ω ) ˆ where Sˆ u u ( ∆x , ωi ) = u 2 i ⋅ Co u u ( ∆x , ωi )

ωi ⋅ ∆x   ˆ Co u u ( ∆x , ωi ) = exp  −C u y ⋅  V  

σu

S (ω ) ˆ Sˆ w w ( ∆x , ωi ) = w 2 i ⋅ Co w w ( ∆x , ωi )

ωi ⋅ ∆x   ˆ Co w w ( ∆x , ωi ) = exp  −Cw y ⋅  V  

where

σw

then 2 S (ω )  D  S u (ωi ) ′ aθ ) I w  ⋅ w 2 i ⋅ψ w (ωi ) ⋅ψ u (ωi ) + (C L′ a z + B C M  2 B CDa y I u  ⋅ σ u2 σw   2

Jˆ ii2 =

(a

2 y

+ a z2 + aθ2

)

2

where ˆ  Co  ψ u (ω )  1 π π u u ( ∆x , ω ) d x1d x 2 ⋅ ∫∫ sin x1 ⋅ sin x 2 ⋅   = 2 ˆ L L Co ψ w (ω )  π   L exp w w ( ∆x , ω )   ∫  sin L x  d x L This integral has previously been solved in Example 6.1, and thus   û )   ωˆ u 2 1 + exp ( −ω ψ u (ωi ) = 4 ⋅  2 + 2π ⋅ 2 2  ˆ ωû2 + π 2  ωu + π 

where

ωû = Cu y f ⋅

  ˆw )   ωˆw 2 1 + exp ( −ω + 2π ⋅ ψ w (ωi ) = 4 ⋅  2 2 2  ˆ ωˆw2 + π 2  ωw + π 

where

ωˆw = Cw y f ⋅

(

)

(

Thus, S Qˆ Qˆ

i i

 ρB 3 =  2m i 

Cov FFR

i

(xr

)

ωi L exp V

ωi L exp

2

2  V  ˆ  ⋅  ⋅ J ii  is defined. At x r = 0   B ωi  

= 0) =

(

πωi ⋅ S Qˆ Qˆ

)(

i i


)

⋅ T ⋅ βi ( x r = 0 ) ⋅ βTi ( x r = 0 ) ⋅ TT

V


192 where

T ⋅ βi ( x r = 0 ) ⋅ βTi ( x r = 0 ) ⋅ TT =  2  π 6 2 a y   ( E I z )  L           S ym .  

6

π  a yaz   E I z E I y L  π  a z2   L 

6

(E I y )

2

   0 0     4 2   π EIw  π  −a z aθ   E I y  GI t +  0 0 2   L L       2 2 π EIw  π    aθ2    GI t + 0 0   L2   L    0 0  0 4

π  −a y a θ   E I z L 

 π 2E I w  GI t + L2 

As could be expected, at x r = 0 it is only V y , V z and M x that applies, and thus

 σ V yV y   σVz Vz  σ M x M x

   ρ B 3  V 2 πωi = ⋅  ⋅ Jˆ ii ⋅   ω m B 4  1 − κ aei ⋅ ζ i − ζ aei i  i   

(

)(

1

)

2   

3  π  ay   EI z  L    3 π  ⋅  az   EI y L   2  π   π 2E I w aθ    GI t + L2   L  

           

and ρV y V z = 1 , ρV y M x = ρV z M x = −1 . At x r = L / 2

πωi ⋅ S Qˆ Qˆ L L   i i ⋅ T ⋅ βi  x r =  ⋅ βTi Cov FFR  x r =  = i  2  4 1 − κ ae ⋅ ζ i − ζ ae 2  i i

(

)(

)

L T   xr = 2  ⋅ T  

where 0 0  0  0 0   0 L T  L T   T ⋅ βi  x r =  ⋅ βi  x r =  ⋅ T =  π a z2  2 2   S ym . L    

0 0 0   

4

     6  π  −a z a y   E I y E I z  L    4  π 2   a 2y   ( E I z )   L  0 0 0

(EI y )

2

As could be expected, at x r = L / 2 it is only M y and M z that applies, and thus  σ M y M y  ρ B 3  V  πωi  = ⋅  ⋅ Jˆ ii ⋅   4 ω m B σ  M z M z  1 − κ aei ⋅ ζ i − ζ aei i  i  

(

Let us consider the case that

)(

  π 2  a z   E I y    ⋅ L      π 2   a  y  L  E I z    1 2

2

)

L exp = L = 500m , V = 40 m s

and adopt the numerical values given in the following tables:

and

a y   1       a z  = 0.5   a  0.1   θ  


193

B m

D m

mi

ωi

ζi

EI y

EI z

GI t

EIw

Kg/m

Rad/s

%

Nm2

Nm2

Nm2

Nm4

20

4

10 5

0.8

0.5

1011

2 ⋅ 1012

5 ⋅ 1010

2 ⋅ 1012

ρ kg

3

m 1.25

xf

CD

C L′

′ CM

Cu y f

Cw y f

Iu

Iw

0.7

5

1.5

1.5

1.0

0.2

0.1

Lu

xf

Lw

m

m

162

13.5

Thus,

ωû = Cu y f ⋅

ωi L exp

ωˆw = Cw y f ⋅

V

ωi L exp V

= 16

  1 + exp ( −ωû )   ωˆ ⇒ ψ u (ωi ) = 4 ⋅  2 u 2 + 2π 2 ⋅ = 0.24 2  ˆ ωû2 + π 2  ωu + π 

= 10

  1 + exp ( −ωˆw )   ωˆ ⇒ ψ w (ωi ) = 4 ⋅  2 w 2 + 2π 2 ⋅ = 0.37 2  ˆ ωˆw2 + π 2  ωw + π 

(

)

(

)

Let us adopt the typical Kaimal type of turbulence spectra S u (ω )

σ u2

=

1.08

(1 + 1.62ω S u (ωi )

⇒

xf

σ u2

Lu / V

xf

Lu / V

S w (ω )

and

)

53

= 0.206

σ w2

=

σ w2

xf

(1 + 2.25ω

S w (ωi )

and

1.5

Lw / V xf

Lw / V

)

53

= 0.229

The joint acceptance is then Jˆ ii2 =

2 2 S (ω )  D  S u (ωi ) ′ aθ ) I w  w 2 i ψ w (ωi ) ψ u (ωi ) + (C L′ a z + B C M  2 B CDa y I u  σ u2 σw  

(a

2 y

+ a z2 + aθ2

)

2

= 0.127

Let us for simplicity also adopt quasi-steady values to the aerodynamic derivatives:  P1*    V ⇒  H 1*  = − B ωi  *  A1  Thus,

κ aei =

ρB

2

2m i

2 ( D / B ) C D   0.7    12.5  ′ C = − L       3.75  ′ C M  

∫

⋅

L exp

(φ φθ B H z

∫(

L

φ y2

* 3

+ φz2

and

)

+ φθ2 B 2 A 3* d x 2

)

+ φθ d x

=

 H 3*   V 2  C L′  31.25  =  * =   ′  9.375   A 3   B ωi  C M

ρ B 2 a z aθ B H 3* + aθ2 B 2 A 3* 2m i

⋅

a 2y + a z2 + aθ2

= 0.136


194

ζ aei =

ρB

2

4m i

∫

⋅

L exp

(φ P

2 * y 1

)

+ φz2 H 1* + φθ φz B A1* d x

∫ (φy + φz 2

2

)

+ φθ2 d x

L

=

2 * 2 * * ρ B 2 a y P1 + a z H 1 + aθ a z B A1

4m i

⋅

a 2y + a z2 + aθ2

= −0.0069

The following is then obtained at x r = 0

 σ V yV y   σVz Vz  σ M x M x

  ρB 3 =  4m i 

 V  ˆ  πωi ⋅  ⋅ J ii ⋅  B ω κ 1 − i   aei ⋅ ζ i − ζ aei 

1

2

(

)(

)

2   

    ⋅     aθ 

    3 π   az   EI y  L   2 π 2 E I w  π     L   GI t + L2      3

π  ay   EI z L 

 σ V y V y ( x r = 0 )   154 kN    ⇒  σ V z V z ( x r = 0 )  =  3.8 kN    σ M x M x ( x r = 0 )  61 kNm  The following is obtained at x r = L / 2 : σ M y M y  ρ B 3  = σ M z M z  4 m i

 V  ˆ  πωi ⋅  ⋅ J ii ⋅  B ω 1 − κ aei ⋅ ζ i − ζ aei i   

(

)(

  π 2  a z   E I y    ⋅ L      π 2   a E I z  y  L    1 2

2

)

σ M y M y ( x r = L / 2 )   612 kNm  = ⇒  24480 kNm    σ M z M z ( x r = L / 2 )  

Chapter 8 MOTION INDUCED INSTABILITIES 8.1 Introduction Static as well as dynamic structural response will in general increase with increasing mean wind velocity. In some cases the response may develop towards what is perceived as unstable behaviour, i.e. the response is rapidly increasing for even a small increase of the mean wind velocity, as indicated in Fig. 6.3. It is seen from Eqs. 6.48 and 6.49 (see also Eqs. 4.69 and 4.82) that in the limit the structural displacement response will become infinitely large if the absolute value of the determinant to the non-dimensional N m od by N m od impedance matrix 2     1  1    ˆ Eη (ω , V ) = I − κ ae −  ω ⋅ d iag    + 2iω ⋅ d iag   ⋅ ( ζ − ζ ae )     ωi    ωi    

(8.1)

is zero. Thus, any stability limit may be revealed by studying the properties of the impedance matrix. Obviously, unstable behaviour is caused by the effects of κ ae and ζ ae . The effects of ζ ae is to change the damping properties of the combined structure

and flow system, while the effects of κ ae is to change the stiffness properties. While we in the entire chapter 6 ignored any motion induced changes to resonance frequencies (defined as the frequency positions of the apexes of the modal frequency response function) this can not be accepted in the search for any relevant instability limit. The reason is explained in chapter 5.2, and as shown in Eq. 5.24, it involves taking into account that the aerodynamic derivatives are modal quantities that have been normalised by and are functions of the mean wind velocity dependent resonance frequencies. Thus (see Eqs. 5.24, 6.51 and 6.52) the content of % % $ $     κ aeij and ζ ae =  ζ aeij (8.2) κ ae =        % % $ $

8 MOTION INDUCED INSTABILITIES

196

are now given by

κ aeij

K aeij

2

∫

(φ

L

ζ aeij

∫ ωi C aeij ρ B 2 ωi (V ) L exp = = ⋅ ⋅ 2 ωi2 M i 4 m i ωi

(

(8.3)

)

(φ

)


T i

∫ ( φi

T

L

)

⋅ Kˆ ae ⋅ φ j d x

T i

 ω (V )  L exp = 2 = ⋅ i  ⋅ T m 2 ωi M i i  ωi  ∫ φi ⋅ φi d x

ρB

2

)

⋅ φi d x

(8.4)

where ωi (V ) is the mean wind velocity dependent resonance frequency associated with mode i and ωi = ωi (V = 0 ) (or as calculated in vacuum). The solution to Eq. 8.1 is an eigen-value problem with N m od roots. Each of these eigen-values represents a limiting behaviour in which the structural response is nominally infinitely large (or irrelevant). I.e., the condition

(

det Eˆ η (ω , V

))

=0

(8.5)

will formally reveal N m od stability limits associated with all the relevant mode shapes

( )

contained in Eˆ η , static or dynamic. In general det Eˆ η will contain complex quantities,

( )

and therefore det Eˆ η = 0 implies the simultaneous conditions that

( ( )) = 0

Re det Eˆ η

and

( ( )) = 0

Im det Eˆ η

(8.6)

As shown above, Eˆ η is a function of the frequency and of the mean wind velocity, and thus, each root will contain a pair of ω and V values which may be used to identify the relevant stability problem. For a static stability limit ω = 0 , and thus, such a limit may simply be identified by a critical wind velocity V cr . For a dynamic stability limit the response is narrow-banded and centred on an in-wind preference or resonance frequency

8.1 INTRODUCTION

197

associated with a certain mode or combination of modes. Thus, the outcome of the eigen-value solution to Eq. 8.5 will identify a dynamic stability limit by a critical velocity V cr and the corresponding in-wind preference or resonance frequency ωr . Of all the eigen-values that may be extracted from Eq. 8.5 the main focus is on the one that represents the stability limit at the lowest mean wind velocity, i.e. it is the lowest V cr (and corresponding ωr ) that has priority. Cases of structural behaviour close to a stability limit may in general be classified according to the response type of displacement that develops. The problem of identification is greatly simplified if the impedance is taken directly from the characteristic behaviour of each stability problem as known from full scale or experimental observations. For a bridge section there are four types of such behaviour. First, there is the possibility of a static type of unstable behaviour in torsion, called static divergence. Second, there is the possibility of a dynamic type of unstable behaviour in the across wind vertical (z) direction, called galloping. Third, there is a possible unstable type of dynamic response in pure torsion, and finally, there is the possibility of an unstable type of dynamic response in a combined motion of vertical displacements and torsion, called flutter. Thus, it is always either rz , rθ or both that are the critical response quantities. It is then only necessary to search for the instability limits associated with the two most onerous modes, φ1 and φ2 with corresponding eigen-frequencies ω1 and ω2 , of which one contain a predominant φz component and the other contain a predominant φθ component. Therefore, the impedance matrix may be reduced to

 1 0 κ ae   11 Eˆ η (ωr ,V cr ) =    − κ 0 1   ae21  

κ ae12  (ωr / ω1 )2 − κ ae22   0 

  (ωr / ω2 )2  0

(8.7)  ζ ζ ζ − −   ae11 ae12 1   +2i   ⋅ 0 / ω ω ζ ζ ζ − −  r 2  ae21 ae22  2    Where (see Eqs. 8.3 and 8.4) κ aeij  ωi (V ) 2  * * * * * =   ∫ φ y φ y P4 + φz i φ y j H 6 + φθi φ y j B A 6 + φ yi φz j P6 + φz i φz j H 4 ρ B 2  ωi   L exp i j  2m i ωr / ω1

0

(

)

+φθi φz j B A 4* +φ yi φθ j B P3* + φz i φθ j B H 3* +φθi φθ j B 2 A 3* d x  

  2 2 2  ∫ φ yi + φz i + φθi d x   L  (8.8)

(

)


198 ζ aeij ρB 2 4 m i

=

ωi (V )  ⋅ ∫ φ yi φ y j P1* + φz i φ y j H 5* + φθi φ y j B A 5* + φ yi φz j P5* + φz i φz j H 1* L ωi 

(

exp

)

+φθi φz j B A1* +φ yi φθ j B P2* + φz i φθ j B H 2* +φθi φθ j B 2 A 2* d x  

and

  2 2 2  ∫ φ yi + φz i + φθi d x   L  (8.9)

(

)

i  = 1,2 . The problem is further simplified if j

φ1 ( x ) ≈ [0 φz

0 ]1

T

φ2 ( x ) ≈ [0 0 φθ ]2

T

   

(8.10)

with corresponding eigen-frequencies ω1 = ωz and ω2 = ωθ , modal eigen-damping ratios ζ 1 = ζ z and ζ 2 = ζ θ , and with modal mass properties m 1 = m z and m 2 = m θ . In that particular case

  1 0  κ aezz Eˆ η (ωr ,V cr ) =   −  0 1  κ aeθ z ω / ω +2i  r z  0

κ aezθ  (ωr / ωz )2 − κ aeθθ   0 

 ζ z − ζ aezz ⋅ ωr / ωθ   −ζ aeθ z 0

  (ωr / ωθ )2 

−ζ aezθ

ζ θ − ζ aeθθ

0

(8.11)

    

and

κ aezz

2

ρ B  ω z (V )  * =   H4 2m z  ωz  2

∫

φz2 d x

L exp 2 ∫ φz d x

2

κ aezθ

ρ B 3  ωz (V )  * =   H3 2m z  ωz 

∫

φz φθ d x

L exp

∫ φz d x 2

L

L

(8.12)

κ aeθθ

ρ B 4  ωθ (V =  2m θ  ωθ

)

2

 

∫

2

φθ d x

L exp A 3* 2

∫ φθ d x

2

κ aeθ z

ρ B 3  ωθ (V )  * =   A4 2m θ  ωθ 

∫

φθ φz d x

L exp

∫ φθ d x 2

L

L

(8.13)

ζ aezz

ρ B 2 ωz (V ) * H1 = 4 m z ωz

∫

φz2 d x

L exp

∫ φz d x 2

L

ζ aezθ

ρ B 3 ω z (V ) * H2 = 4 m z ωz

∫

φz φθ d x

L exp

∫ φz d x 2

L

(8.14)

8.2 STATIC DIVERGENCE

ζ aeθθ

ρ B ωθ (V ) * A2 = 4 m θ ωθ 4

∫

φθ2 d x

L exp

ζ aeθ z

∫ φθ d x 2

199

ρ B 3 ωθ (V ) * A1 = 4 m θ ωθ

∫

φθ φz d x

L exp

∫ φθ d x 2

L

L

(8.15) where ωz (V ) and ωθ (V ) are the mean wind velocity dependent resonance frequencies associated with φ1 ( x ) ≈ [0 φz 0 ]1 and φ2 ( x ) ≈ [0 0 φθ ]2 . A purely single mode T

T

unstable behaviour contains motion either in the vertical direction (i.e. galloping) or in torsion. Such an instability limit may then be identified from the first or the second row of the matrices in Eq. 8.11 alone, in which case ωr = ωz (V cr ) or ωr = ωθ (V cr ) . Otherwise, the unstable behaviour contains a combined motion in the vertical direction and torsion (i.e. flutter), in which case the instability limit may be identified from Eq. 8.11, and ωr = ωz (V cr ) = ωθ (V cr ) . Motion induced coupling effects between rz and rθ (i.e. flutter) will only occur if the off–diagonal terms in Eq. 8.11 are unequal to zero, i.e. if ∫ φz φθ d x ≠ 0 (see Eqs. 8.12 – 8.15). L exp

8.2

Static divergence

Let φ2 be the mode shape in predominantly torsion that has the lowest eigen–frequency. Let us for simplicity assume that

φ2 ≈ [0 0 φθ ]

T

(8.16)

At ωr = 0 , the instability effect is static and not dynamic. It is simply a problem of loosing torsion stiffness due to interaction effects with the air flow. Thus, the impedance in Eq. 8.11 is reduced to Eˆη (ωr = 0,V cr ) = 1 − κ aeθθ

where:

κ aeθθ =

2

ρ B  ωθ (V cr )  *   A 3 2m θ  ωθ  4

(8.17)

∫

φθ2 d x

L exp

∫ φθ d x 2

L

It is seen that Eˆη (ωr = 0, V cr ) = 0 when κ aeθθ = 1 . Thus, a static divergence type of

instability limit may be identified under the condition that


200

2

ρ B  ωθ (V cr )  *   A 3 2m θ  ωθ  4

∫

φθ2 d x

L exp

=1

∫ φθ d x 2

(8.18)

L

Since this is a purely static type of unstable behaviour the quasi-static version of A 3* from Eq. 5.26 applies, and thus, the following critical mean wind velocity for static divergence is obtained

V cr

 2 ∫ φθ d x  θ 2 m = B ⋅ ωθ ⋅  ⋅ L 2  ρ B 4C M ′ ∫ φθ d x  L exp 

1/2

     

(8.19)

8.3 Galloping Let φ1 be the mode shape with the lowest eigen-frequency ω1 = ωz whose main component is φz , i.e.

φ1 ≈ [0 φz

0]

T

(8.20)

Since the resonance frequency associated with this mode is ωz (V ) , then ωr = ωz (V cr )

(8.21)

and the impedance in Eq. 8.11 is reduced to

(

)

2 Eˆη (ωr , V cr ) = 1 − κ aezz − (ωr / ωz ) + 2i ζ z − ζ aezz ωr / ωz

(8.22)

where 2

κ aezz =

ρ B 2  ωr  *   H4 2m z  ωz 

∫

φz2 d x

L exp

∫

L

φz2 d x

and

ζ aezz =

ρ B 2 ωr * H1 4 m z ωz

∫

φz2 d x

L exp

∫ φz d x 2

L

8.4 DYNAMIC STABILITY LIMIT IN TORSION

201

Setting the real and imaginary parts of Eq. 8.22 equal to zero, a dynamic stability limit may then be identified at an in-wind resonance frequency

  ρB 2 * ωr = ωz 1 + H4  2m z  

φz2 d x 

∫

−1 / 2

  2 ∫ φz d x  L 

L exp

(8.23)

when the damping properties are such that

ζ z = ζ aezz =

∫

ρ B ωr * H1 4 m z ωz 2

φz2 d x

L exp

(8.24)

∫ φz d x 2

L

This type of stability problem is called galloping. It is seen that a galloping instability can only occur if H 1* attains positive values. (For a flat plate H 1* is consistently negative, see Fig. 5.3, but this is a property that vanishes for cross sections with increasing bluffness.) Adopting the quasi-static versions of the aerodynamic derivatives given in Eq. 5.26, then the stability limit is defined by the following mean wind velocity

∫ φz d x 2

V cr = B ωz ⋅

(

ζz

− C L′ + C D ⋅ D / B

)

⋅

4 m z

ρB 2

⋅

L

∫

φz2 d x

(8.25)

L exp

An analytical solution to the problem of galloping was first presented by den Hartog [29], showing that galloping can only occur if C L′ < −C D ⋅ D / B .

8.4 Dynamic stability limit in torsion A stability problem in torsion is related to galloping in the sense that it involves a single mode type of motion. Let φ2 be the mode shape with the lowest eigen-frequency ω2 = ωθ whose main component is φθ , i.e.

φ2 ≈ [0 0 φθ ]

T

Since the resonance frequency associated with this mode is ωθ (V ) , then

(8.26)


202

ωr = ωθ (V cr )

(8.27)

and the impedance in Eq. 8.11 is reduced to

(

)

2 Eˆη (ωr , V cr ) = 1 − κ aeθθ − (ωr / ωθ ) + 2i ζ θ − ζ aeθθ ωr / ωθ

(8.28)

where

κ aeθθ =

2

ρ B  ωr  *   A3 2m θ  ωθ  4

∫

φθ2 d x

L exp

∫ φθ d x 2

and

ζ aeθθ =

ρ B ωr * A2 4 m θ ωθ

L

4

∫

φθ2 d x

L exp

∫ φθ d x 2

L

Setting the real and imaginary parts of Eq. 8.28 equal to zero, a dynamic stability limit may then be identified at an in-wind resonance frequency

 φθ2 d x ∫  L exp ρB 4 ωr = ωθ 1 + ⋅ A 3* ⋅ 2  2m θ ∫ φθ d x  L 

     

−1 / 2

(8.29)

when the damping properties are such that

ζ θ = ζ aeθθ =

ρ B ωr * A2 4 m θ ωθ 4

∫

φθ2 d x

L exp

∫ φθ d x 2

(8.30)

L

It is seen that an instability in pure torsion can only occur if A 2* attains positive values. (For a flat plate A 2* is consistently negative, see Fig. 5.3.) Since the quasi-static value of A 2* is zero, it is futile to define a stability limit based on the quasi-static theory.

8.5 FLUTTER

8.5

203

Flutter

As mentioned above, flutter is a dynamic stability problem where rz couples with rθ . Such coupling occurs via the off-diagonal terms κ aezθ and κ aeθ z in Eq. 8.11 above, and therefore, it is most prone to occur between modes φ1 and φ2 that are shape-wise similar and whose main components are φz and φθ . Experimental observations show that it is usually the aerodynamic forces associated with the motion in torsion that are the driving forces in the coupling process. Let φ2 be the mode shape with the lowest eigen-frequency ω2 = ωθ whose main component is φθ , i.e.

φ2 ≈ [0 0 φθ ]

T

(8.31)

Let φ1 be another mode that shape-wise is similar to φ2 and whose main component is

φz , i.e. φ1 ≈ [0 φz

0]

T

(8.32)

and whose eigen–frequency is ω1 = ωz . A flutter stability limit is then identified by

(

)

det Eˆ η (ωr , V cr ) = 0 where Eˆ η (ωr , V cr ) is given in Eq. 8.11. Since rz couples with rθ

into a joint resonant motion, then ωr = ωz (V cr ) = ωθ (V cr )

(8.33)

From a computational point of view it is convenient to split Eˆ η into four parts, i.e.

(

Eˆ η = Eˆ 1 + Eˆ 2 + 2i Eˆ 3 + Eˆ 4

)

(8.34)

where 2 1 − κ 0 aezz − ( ωr / ωz ) ˆ   E1 =   − κ 0 aeθ z    ζ z − ζ ae ⋅ ωr / ωz 0  zz  Eˆ 3 =   −ζ ae ⋅ ωr / ωθ  0 θz  

(

)

−κ aezθ 0   Eˆ 2 =  2 0 1 − κ ae − (ωr / ωθ )  θθ   −ζ aezθ ⋅ ωr / ωz  0  Eˆ 4 =  0 ζ θ − ζ ae ⋅ ωr / ωθ  θθ  

(

)

     (8.35)    


204

The stability limit is then defined by the following two conditions

( ( )) = det (Eˆ

Re det Eˆ η

)

(

)

+ Eˆ 2 − 4 ⋅ det Eˆ 3 + Eˆ 4 = 0

1

( ( )) = 2 ⋅ det (Eˆ

Im det Eˆ η

1

)

(

(8.36)

)

+ Eˆ 4 + det Eˆ 2 + Eˆ 3  = 0 

(8.37)

Fully expanded these equations become

( ( )) = 1 − κ

Re det Eˆ η

zz

− κθθ + κ zz ⋅ κθθ − κ zθ ⋅ κθ z

−4 ⋅ (ζ z − ζ zz ) ⋅ (ζ θ − ζ θθ ) − ζ zθ ⋅ ζ θ z  ⋅ (ωr / ωz ) ⋅ (ωr / ωθ ) − (1 − κθθ ) ⋅ (ωr / ωz ) − (1 − κ zz ) ⋅ (ωr / ωθ ) + (ωr / ωz ) ⋅ (ωr / ωθ ) 2

2

2

2

=0

( ( )) = 2 ⋅ {(1 − κ

Im det Eˆ η

(8.38)

) ⋅ (ζ z

θθ

− ζ zz ) − κθ z ⋅ ζ zθ  ⋅ ωr / ωz

+ (1 − κ zz ) ⋅ (ζ θ − ζ θθ ) − κ zθ ⋅ ζ θ z  ⋅ ωr / ωθ − (ζ θ − ζ θθ ) ⋅ (ωr / ωθ ) ⋅ (ωr / ωz ) − (ζ z − ζ zz ) ⋅ (ωr / ωz ) ⋅ (ωr / ωθ ) 2

2

}

=0 (8.39) where (see Eqs. 8.12 – 8.15)

κ aezz

2

ρ B  ωr  * =   H4 2m z  ωz  2

∫

φz2 d x

L exp

∫ φz d x 2

κ aezθ

∫

2

ρ B  ωr  * =   H3 2m z  ωz  3

∫ φz d x 2

L

κ aeθθ

2

ρ B  ωr  * =   A3 2m θ  ωθ  4

∫

L

2

φθ d x

L exp

∫ φθ d x 2

κ aeθ z

2

ρ B  ωr  * =   A4 2m θ  ωθ  3

ζ aezz =

ζ aeθθ

∫

ρ B 4 ωr * A2 = 4 m θ ωθ

∫

∫

∫ φθ d x 2

L

ζ aezθ =

φθ2 d x

L exp

φθ φz d x

∫ φθ d x 2

L

φz2 d x

ρ B ωr * L exp H1 4 m z ωz φz2 d x L

∫

L exp

L 2

φz φθ d x

L exp

ζ aeθ z

∫

φz φθ d x L ρ B ωr * exp H2 4 m z ωz φz2 d x L 3

ρ B 3 ωr * A1 = 4 m θ ωθ

∫

∫

φθ φz d x

L exp

∫ φθ d x 2

L

8.5 FLUTTER

205

The solution procedure demands iterations, because the aerodynamic derivatives can only be read off if the outcome, ωr and V cr , are known. The theory of flutter was first presented by Theodorsen [28]. In cases where ωθ / ωz is larger than about 1.5, then

Selberg’s formula [22] may be used to provide a first estimate of the mean wind velocity that defines the flutter stability limit 1/2

V cr

   ω 2  ( m ⋅ m )1 / 2    θ = 0.6 B ωθ ⋅  1 −  z   ⋅ z  3 ρB    ωθ   

(8.40)

Example 8.1: Let us consider a slender horizontal beam type of bridge with a cross section whose aerodynamic properties are close to those of an ideal flat plate, and set out to calculate the possible stability limits associated with the two mode shapes φ1 = [0 φz

0]

φ2 = [0 0 φθ ]

T

T

with corresponding eigen-frequencies ωz and ωθ , and with modally equivalent and evenly distributed masses m z and m θ . It is for simplicity assumed that φz ≈ φθ and that L exp = L . Let us allot the following values to the necessary structural quantities

ρ (kg/m3)

B (m)

1.25

20

m z (kg/m) 10

4

m θ (kgm2/m) 6 ⋅ 10

ωz (rad/s)

ζ z = ζθ

0.8

0.005

5

We wish to investigate the properties of the instability limits at various values of the frequency ratio ωθ ωz , and thus it is assumed that ωz = 0.8 rad/s while ωθ is arbitrary between

ωθ ωz = 1 and ωθ ωz = 3 . To simplify the relevant expressions, let us introduce the following notation:

βz =

ρB 2 m z

= 0.05

βθ =

ρB 4 m θ

= 0.33

γ =

ωθ ωz

and

ωˆ r =

ωr ωθ

Due to the flat plate type of aerodynamic properties it is in this particular case only static divergence and flutter that may occur. The flat plate aerodynamic derivatives are given in Eq. 5.27 (and shown in Fig. 5.3), i.e.:


206

 H 1*  * H 2  * H 3  * H 4

A1*   A 2*   A 3*   A 4* 

 −2π FVˆ    π 1 + F + 4 GVˆ Vˆ 2 =  2π FVˆ − G 4 Vˆ    π 1 + 4 GVˆ 2 

(

)

(

−

) )

(

π

   1 − F − 4 GVˆ Vˆ   FVˆ − G 4 Vˆ    π ˆ  GV 2  −

π

2

FVˆ

( π ( 2

8

)

)

where Vˆ = V  B ωi (V )  is the reduced velocity, and where  ωˆ F i  2

 J 1 ⋅ ( J 1 + Y 0 ) + Y1 ⋅ (Y1 − J 0 ) =  ( J 1 + Y 0 )2 + (Y1 − J 0 )2

 ωˆ G i  2

and

J 1 ⋅ J 0 + Y1 ⋅ Y 0  =−  ( J 1 + Y 0 )2 + (Y1 − J 0 )2

are the real and imaginary parts of the so-called Theodorsen’s circulatory function. J n (ωî 2 ) and Y n (ωî 2 ) , n = 0 or 1 , are first and second kind of Bessel functions with order n . ωˆ i is the non-

dimensional resonance frequency, i.e. ωˆ i = B ωi (V ) / V . For an ideal flat plate type of cross ′ = π 2 (see quasi static solution given in Eq. 5.29). Thus, the stability limit with section C M respect to static divergence is identified by 12

 2m θ 1  V cr = ⋅  ′  B ωθ  ρ B 4 C M

⇒

V cr = B ωθ

2

βθ π

≈ 1.96

With respect to the flutter stability limit an approximate solution can be obtained from Eq. 8.40 (the Selberg formula), rendering 1/2

2 1/2   V cr ω   ( m ⋅ m )   = 0.6 ⋅  1 −  z   ⋅ z θ3  B ωθ ρB    ωθ   

≈ 1.67 ⋅ 1 − γ −2

An exact solution can only be obtained from the simultaneous solution of Eqs. 8.38 and 8.39. Introducing the simplifications that L exp = L and φz ≈ φθ and the abbreviations for β z , βθ , γ and ωˆ r defined above, then 2

κ aezz =

βz * 2 2 ρ B 2  ωr  * H 4 γ ωˆr   H4 = 2m z  ωz  2

κ aeθθ =

2

κ aezθ =

βz ρ B 3  ωr  * B H 3*γ 2 ωˆ r2   H3 = 2m z  ωz  2

ρ B 4  ωr  * βθ * 2 A 3 ωˆ r   A3 = 2m θ  ωθ  2

κ aeθ z =

ρ B 3  ωr  * βθ 1 * 2 A 4 ωˆ 2   A4 = 2m θ  ωθ  2 B

ζ aezz =

ρ B 2 ωr * β z * H1 = H 1 γωˆ r 4 m z ωz 4

ζ aezθ =

ρ B 3 ωr * β z H2 = B H 2*γωˆ r 4 m z ωz 4

ζ aeθθ =

ρ B 4 ωr * βθ * A2 = A 2 ωˆ r 4 m θ ωθ 4

ζ aeθ z =

ρ B 3 ωr * βθ 1 * A1 = A1 ωˆ r 4 m θ ωθ 4 B

2

2

8.5 FLUTTER

207

Thus, Eqs. 8.29 and 8.30 are reduced to

( ( )) = 1 − 1 + γ

Re det Eˆ η

2

+ 4 γζ z ζ θ +

β β β β  +γ 2 1 + z H 4* + θ A 3* + z θ 2 2 4 

βz 2

(

( ( )) = 2ωˆ ζ γ + ζ

Im det Eˆ η

r

z

βθ

(

)

 A 3*  ωˆ r2 + γ ζ θ β z γ H 1* + ζ z βθ A 2* ωˆ r3   A1* H 2* − A 2* H 1* + A 3* H 4* − A 4* H 3*  ωˆ r4 = 0 

γ 2 H 4* +

2

)

θ

−

(

)

1 β z γ 2 H 1* + βθ A 2* ωˆ r 4

 β  β  − ζ z  θ A 3* + γ  + ζ θ γ 2  z H 4* + 1   ωˆ r2 2 2      1 β β   +γ 2  z θ H 1* A 3* − H 2* A 4* − H 3* A1* + H 4* A 2* + β z H 1* + βθ A 2*  ωˆ r3  = 0 4  8  

(

) (

)

It is seen that the solution of these equations requires the search for the lowest identical roots in a fourth and a third degree polynomial. Adopting ideal flat plate aerodynamic derivatives (see expressions above) the solution is shown in the upper diagram in Fig. 8.2 (together with the approximate solution given by Selberg’s formula). The corresponding values of ωˆ r are shown in

( ( ))

the lower diagram in Fig. 8.2. At ωθ / ωz = 2 the development of Im det Eˆ η

( ( )) with increasing values of V / ( B ω ) is shown in Fig. 8.1.

Re det Eˆ η

θ

Fig. 8.1

Development of imaginary and real parts at increasing values of V / ( B ωθ )

and

Appendix A TIME DOMAIN SIMULATIONS A.1

Introduction

It is in the following taken for granted that the stochastic space and time domain simulation of a process x implies the extraction of single point or simultaneous multiple point time series from known frequency domain cross spectral information about the process. The process may contain coherent or non-coherent properties in space and time. Thus, a multiple point representation is associated with the spatial occurrence of the process. For a non-coherent process there is no statistical connection between the simulated time series that occur at various positions in space, and thus, the simulation may be treated as a representation of independent single point time series. This type of simulation is shown in chapter A.2. For a coherent process there is a prescribed statistical connection between each of the spatial representatives within a set of M simulated time series. E.g., if the simulated time series represent the space and time distribution of a wind field, there will be a certain statistical connection between the instantaneous values x m (t ) , m = 1,2,...., M that matches the spatial properties of the wind field. Such a simulation is shown in chapter A.3. The simulation procedure presented below is taken from Shinozuka [23] and Deodatis [24]. Simulating time series from spectra is particularly useful for two reasons. First, there are some response calculations that render results which are more or less narrow banded (or contain beating effects), and thus, they do not necessarily comply with the assumptions behind the peak factor given in Eq. 2.45. These cases may require separate time domain simulations to establish an appropriate peak factor for the calculation of maximum response. This application will usually only require single point simulations. Secondly, if the relevant cross spectra of the wind field properties in frequency domain are known, there is always the possibility of a time domain simulation of the entire wind field, or those of the flow components that are deemed necessary. Together with the buffeting load theory in chapter 5.1 this is a tempting option, as time domain step-wise load effect integration may be performed, and thus, the response calculation may be carried out in time domain instead of the frequency domain approach that is shown in chapter 6. The mathematical procedure for such an approach may be found in many text books, see e.g. Hughes [25]. The main advantage is that such an approach may contain many of the non-linear effects that had to be simplified or discarded in the linear theory that was required for a frequency domain solution. The disadvantage is that motion induced load effects can only be fully included if a new set of indicial functions are introduced (see e.g. Scanlan [26]). These may not be readily available.

A.2 SIMULATION OF SINGLE POINT TIME SERIES

x (t ) =

N

∑ ck cos (ωk t + ψ k )

(A.2)

k =1

1/2

ck = 2 ⋅ S x (ωk ) ⋅ ∆ωk 

where

211

(A.3)

and where ψ k are arbitrary phase angles between zero and 2π , one for each harmonic component. Alternatively, Eq. A.2 may be replaced by the exponential format (often encountered in the literature)

 N  x (t ) = Re  ∑ ck ⋅ exp i (ωk t + ψ k )   k =1  The variance of x (t ) is

N

c2

∑ 2k

(A.4)

, which in the limit of ∆ω → 0 and N → ∞ ,

k =1

N

c2

∞

∑ k = ∫ S x (ω ) d ω ∆ω →0 k =1 2

σ x2 = lim

N →∞

(A.5)

0

I.e., if the discretization is sufficiently fine, then the variance of the simulated representative, x (t ) , is equal to or close enough to the variance of the parent variable. The procedure is further illustrated in Example A.1 and Fig. A.2. Any number of such representatives may be simulated simply by changing the choice of phase angles. Obviously, the accuracy of such a simulation depends on the discretization fineness, but there is also the unfavourable possibility of aliasing. Let ωc be the upper cut-off frequency, beyond which there is none or only negligible spectral information about the process. Assuming constant frequency segments

∆ω = ωc / N

(A.6)

then each simulated time series will be periodic with period T = 2π / ∆ω

(A.7)

Thus, time series without aliasing will be obtained if they are generated with a time step

∆t ≤ 2π / ( 2ωc )

(A.8)

A.3 SIMULATION OF NON-COHERENT TIME SERIES

213

Example A.1: The top diagram in Fig. A.2 shows the single point single sided spectrum of a process x of which we wish to portray two representatives in time domain. As shown, the frequency span of the spectrum is first divided into five equal frequency segments, and the corresponding values ωk and S x (ωk ) , k = 1, 2,...,5 , are read off. Thus the process is represented by five harmonic

components whose amplitudes ck = 2 ⋅ S x (ωk ) ⋅ ∆ω are given in the far right hand side column in the table of Fig. A.2. Thus

x (t ) =

5

∑

k =1

2 S x (ωk ) ∆ω ⋅ cos (ωk t + ψ k )

What then remains is to choose five arbitrary value of ψ k . In Fig. A.2 the five cosine components are first shown by fully drawn lines, representing a certain choice of ψ k values. The sum of these components shown in the lower diagram in Fig. A.2 is an arbitrary representation of the process x (t ) . If the second and the fourth of these components are moved an arbitrary time shift, then together with the remaining unchanged components they sum up to become another arbitrary representation of the process shown by the broken line in Fig. A.2. As can be seen, the two simulated representatives look quite different in time domain, although they come from the same spectral density. What is important is that they both have zero mean and the same variance, i.e. they have identical statistical properties up to and including the variance.

A.3

Simulation of spatially non–coherent time series

While the procedure presented above may be used to simulate single point time series representatives of x , it is not applicable if we wish to simulate multiple point time series whose properties are expected to be distributed according to certain coherence properties. E.g., let us assume that we wish to simulate the turbulence components u  x y f , z f ,t x =v (A.9) w 

(

)

of a stationary and homogeneous wind field at a chosen number of points M in a plane perpendicular to the main flow direction. It is then important to capture the fact that these time series are representatives of simultaneous events, and therefore, they must contain the appropriate spatial coherence properties that are characteristic to the process. For simplicity it is in the following assumed that cross spectra between the u , v and w components are negligible, i.e. that x S xy (ω , ∆s ) ≈ 0 (A.10)  = u ,v ,w y

A. TIME DOMAIN SIMULATIONS

214

where ∆s is the spatial separation in the y f − z f plane. We will then only need information about the cross spectra of the turbulence components themselves, S xx (ω , ∆s ) . Let Cov x m x n (τ ) be the covariance and S x m x n (ω ) the corresponding cross spectral density between two arbitrary points m and n . As shown in chapter 2.6 these quantities constitute a Fourier transform pair. An M by M cross spectral density matrix

Sxx

 S x1 x1   (ω ) =  S xm x1   S x M x1

S x1 x n S xm x n S x M xn

S x1 x M    S xm x M     S x M x M 

(A.11)

will then contain all the space and frequency domain information that is necessary for a time domain simulation of M time series with the correct statistical properties for a special representation of the process. It follows from the assumptions of stationarity and homogeneity that

and thus,

Cov x m x n = Cov x n x m

(A.12)

S x m x n = S x*n x m

(A.13)

This implies that Sxx (ω ) is Hermitian and non–negative definite. A Cholesky decomposition of Sxx will then render a lower triangular matrix

0  G x1 x1  Gx2 x2  G x 2 x1  Gxx (ω ) =   Gx x G xm x 2  m 1   G x M x1 G x M x 2 whose properties are such that

0

0

0

0

0

0

0

0

G xm x n

G xm xm

Gx M xn

G x M xm

0

0 Gx M x M

Sxx (ω ) = Gxx ⋅ G*xxT Assuming a frequency segmentation of N equidistant points, the simulated simultaneous time series at m = 1, 2,...., M are then given by

         

(A.14)

(A.15)


x m (t ) =

m

N

∑ ∑ Gm n ( ω j ) ⋅

n =1 j =1

(

2 ∆ω ⋅ cos ω j ⋅ t + ψ n j

215

)

(A.16)

where j is the frequency segment number and ψ n j is an arbitrary phase angle between zero and 2π . In most cases of a homogeneous wind field (see Eq. 2.87) S (ω , ∆s ) = S (ω ) ⋅ Sˆ (ω , ∆s ) xx

x

xx

(A.17)

where S x is the single-point spectral density of the process, ∆s = s x m − s x n is the spatial separation between points x m and x n , and where

Sˆ xx (ω , ∆s ) = Coh xx (ω , ∆s ) ⋅ exp iϕ xx (ω ) 

(A.18)

ˆ ⋅G ˆ *T , then the time series at Thus, defining a Cholesky decomposition Sˆ xx (ω ) = G xx xx m = 1, 2,...., M are given by

x m (t ) =

m

N

∑ ∑ Gˆ m n (ω j ) ⋅

( )

(

2 S x ω j ⋅ ∆ω ⋅ cos ω j ⋅ t + ψ n j

n =1 j =1

)

(A.19)

ˆ where Gˆ m n is the content of G xx (i.e. the reduced versions of Gm n in Eq. A.14)

 Gˆ11 0 0 0 0   Gˆ 21 Gˆ 22 0 0 0  ˆ (ω ) =  G xx  Gˆ Gˆ m 2 Gˆ m n Gˆ m m  m1   ˆ ˆ Gˆ M n Gˆ M m G M 1 G M 2 and where a Cholesky decomposition will render

( )

(

)

0

1/2

Gˆ11 ω j = Sˆ xx ω j ,0   

(

)

(A.20)

(A.21) 1/2

m −1   Gˆ m m ω j = Sˆ xx ω j ,0 − ∑ Gˆ m2 k ω j  k =1  

( )

0   0    0     Gˆ M M 

( )

(A.22)


216

n −1

( )

Gˆ m n ω j =

( ) Gˆ n n (ω j )

( )

Sˆ xx (ω , ∆s ) − ∑ Gˆ m k ω j ⋅ Gˆ n k ω j k =1

(A.23)

Example A.2:

A process x is statistically distributed in time and space. Its cross-spectrum S xx (ω , ∆s ) is defined by the product between the single point spectrum S x (ω ) shown in Fig. A.3 and its rootcoherence function

Coh xx (ω , ∆s ) shown in Fig. A.4. I.e., S xx (ω , ∆s ) = S x (ω ) ⋅ Coh xx (ω , ∆s )

The phase spectrum exp iϕxx (ω )  is assumed equal to unity for all relevant values of ω and ∆ s . Let us set out to simulate the process at three points in space, each a distance 10 m apart. Thus, ∆s = [ ∆s1

∆s2

∆s3 ] = [0 10 20 ] T

T

Let us for simplicity settle with the three point frequency segmentation shown in Fig. A.3. I.e. ω = [ω1 ω2

ω3 ] = [0.3 0.7 1.1 ] T

T

and

∆ω = 0.4

(It should be noted that this frequency segmentation is only justified by the wish of obtaining mathematical expressions with reasonable length, such that a complete solution may be presented. For any practical purposes such a coarse segmentation will most often render unduly inaccurate results.) The single point spectrum at these frequency settings are then (see Fig. A.3) Sx = S x (ω1 ) S x (ω2 ) S x (ω3 )  = [4.0 7.6 3.0 ] T

T

while the corresponding values of the root coherence function are given by (see Fig. A.4)


Fig. A.3

Fig. A.4

Single point spectrum

Root coherence function at ω = 0.3, 0.7 a n d 1.1

217


218 Coh xx (ω , ∆s ) :

0 1.0 1.0 1.0

0.3 0.7 1.1

ω

∆s 10 0.6005 0.3042 0,1541

20 0.3606 0.0926 0.0238

sym .   1   ˆ S ω = ∆ = 0.3, s 0.6005 1 xx j mn   0.3606 0.6005 1 

(

Thus,

)

sym .   1   ˆ S ω = ∆ = 0.7, s 0.3042 1 xx j mn   0.0926 0.3042 1 

(

)

sym .   1   ˆ ω S = ∆ = 1.1, s 0.1541 1 xx j mn   0.0238 0.1541 1 

(

)

Gˆ11 0 0    ˆ ˆ ˆT ˆ ˆ ˆ 0  is defined such that S Gxx ω j = G 21 G 22 xx ω j , ∆sn = Gxx ⋅ Gxx   Gˆ 31 Gˆ 32 Gˆ 33  Its content is given by (see Eqs. A.21 – A.23)

(

( )

( ) ( ) Gˆ 21 (ω j ) = Sˆ xx (ω j , ∆s21 = 10 )

)

12

Gˆ11 ω j = Sˆ xx ω j , ∆s11 = 0   

( )

Gˆ11 ω j ,

( ) ( ) ( ) Gˆ 31 (ω j ) = Sˆ xx (ω j , ∆s31 = 20 ) Gˆ11 (ω j ) Gˆ 32 (ω j ) = Sˆ xx (ω j , ∆s32 = 10 ) − Gˆ 31 (ω j ) ⋅ Gˆ 21 (ω j )    12

2 Gˆ 22 ω j = Sˆ xx ω j , ∆s22 = 0 − Gˆ 21 ωj   

( )

(

)

( )

( )

( )

Gˆ 22 ω j

12

2 2 Gˆ 33 ω j = Sˆ xx ω j , ∆s33 = 0 − Gˆ 31 ω j − Gˆ 32 ωj   

Thus, Gˆ = 1  11  ω1 = 0.3 ⇒ Gˆ 21 = 0.6005 ˆ G31 = 0.3606

   = ( 0.6005 − 0.3606 ⋅ 0.6005 ) 0.7996 = 0.4802    = 1 − 0.3606 2 − 0.4802 2 = 0.7996

Gˆ 22 = 1 − 0.6005 2 = 0.7996 Gˆ 32 Gˆ 33

0 0   1   ˆ ⇒ Gxx (ω1 = 0.3 ) = 0.6005 0.7996 0  0.3606 0.4802 0.7996 


Gˆ = 1  11  ω2 = 0.7 ⇒ Gˆ 21 = 0.3042 ˆ G31 = 0.0926

   = ( 0.3042 − 0.0926 ⋅ 0.3042 ) 0.9526 = 0.2898    = 1 − 0.0926 2 − 0.2898 2 = 0.9526

Gˆ 22 = 1 − 0.3042 2 = 0.9526 Gˆ 32 Gˆ 33

0 0   1  ˆ (ω = 0.7 ) = 0.3042 0.9526 ⇒G 0 xx 2   0.0926 0.2898 0.9526 

Gˆ = 1  11  ω3 = 1.1 ⇒ Gˆ 21 = 0.1541 ˆ G31 = 0.0238

   = ( 0.1541 − 0.0238 ⋅ 0.1541 ) 0.9881 = 0.1522    = 1 − 0.0238 2 − 0.1522 2 = 0.9881

Gˆ 22 = 1 − 0.1541 2 = 0.9881 Gˆ 32 Gˆ 33

0 0   1  ˆ (ω = 1.1 ) =  0.1541 0.9881 ⇒G 0  xx 1  0.0238 0.1522 0.9881 

Denoting  2 S (ω = 0.5 ) ⋅ ∆ω    x 1  a1     2 ⋅ 4 ⋅ 0.4  1.79        a 2  =  2 S x (ω2 = 0.7 ) ⋅ ∆ω  =  2 ⋅ 7.6 ⋅ 0.4  ≈ 2.46     a 3   2 S (ω = 1.1 ) ⋅ ∆ω   2 ⋅ 3 ⋅ 0.4  1.55  x 3  

then the three time series are given by (see Eq. A.19)

x1 (t ) =

1

3

∑ ∑ Gˆ1n (ω j ) ⋅

n =1 j =1

( )

(

2 S x ω j ∆ω ⋅ cos ω jt + ψ n j

)

= Gˆ11 (ω1 ) a1 cos (ω1t + ψ 11 ) + Gˆ11 (ω2 ) a 2 cos (ω2t + ψ 12 ) + Gˆ11 (ω3 ) ⋅ a 3 ⋅ cos (ω3t + ψ 13 ) = 1.79 ⋅ cos ( 0.3t + ψ 11 ) + 2.46 ⋅ cos ( 0.7t + ψ 12 ) + 1.55 ⋅ cos (1.1t + ψ 13 ) x 2 (t ) =

2

3

∑ ∑ Gˆ 2 n (ω j ) ⋅

n =1 j =1

( )

(


)

= Gˆ 21 (ω1 ) a1 cos (ω1t + ψ 11 ) + Gˆ 21 (ω2 ) a 2 cos (ω2t + ψ 12 ) + Gˆ 21 (ω3 ) ⋅ a 3 ⋅ cos (ω3t + ψ 13 ) + Gˆ 22 (ω1 ) a1 cos (ω1t + ψ 21 ) + Gˆ 22 (ω2 ) a 2 cos (ω2t + ψ 22 ) + Gˆ 22 (ω3 ) ⋅ a 3 ⋅ cos (ω3t + ψ 23 ) = 1.075 ⋅ cos ( 0.3t + ψ 11 ) + 0.748 ⋅ cos ( 0.7t + ψ 12 ) + 0.239 ⋅ cos (1.1t + ψ 13 ) +1.431 ⋅ cos ( 0.3t + ψ 21 ) + 2.343 ⋅ cos ( 0.7t + ψ 22 ) + 1.532 ⋅ cos (1.1t + ψ 23 )

219


220 x 3 (t ) =

3

3

∑ ∑ Gˆ 3n (ω j ) ⋅

n =1 j =1

( )

(


)

= Gˆ 31 (ω1 ) a1 cos (ω1t + ψ 11 ) + Gˆ 31 (ω2 ) a 2 cos (ω2t + ψ 12 ) + Gˆ 31 (ω3 ) ⋅ a 3 ⋅ cos (ω3t + ψ 13 ) + Gˆ 32 (ω1 ) a1 cos (ω1t + ψ 21 ) + Gˆ 32 (ω2 ) a 2 cos (ω2t + ψ 22 ) + Gˆ 32 (ω3 ) ⋅ a 3 ⋅ cos (ω3t + ψ 23 ) + Gˆ 33 (ω1 ) a1 cos (ω1t + ψ 31 ) + Gˆ 33 (ω2 ) a 2 cos (ω2t + ψ 32 ) + Gˆ 33 (ω3 ) ⋅ a 3 ⋅ cos (ω3t + ψ 33 ) = 0.646 ⋅ cos ( 0.3t + ψ 11 ) + 0.228 ⋅ cos ( 0.7t + ψ 12 ) + 0.039 ⋅ cos (1.1t + ψ 13 ) +0.86 ⋅ cos ( 0.3t + ψ 21 ) + 0.713 ⋅ cos ( 0.7t + ψ 22 ) + 0.236 ⋅ cos (1.1t + ψ 23 ) +1.431 ⋅ cos ( 0.3t + ψ 31 ) + 2.343 ⋅ cos ( 0.7t + ψ 32 ) + 1.532 ⋅ cos (1.1t + ψ 33 ) What then remains is to ascribe arbitrary values (between 0 and 2π ) to the phase angles, ψ n j . The following is chosen: 0.7 0.6 0.3    ψ = 2π ⋅  0.1 0.4 0.2   0.1 0.7 0.8 

Fig. A.5

Simulated time series

A.4 THE CHOLESKY DECOMPOSITION

221

The simulated time series are shown in Fig. A.5 ( T = 600 s and ∆t = 0.06 s). The standard deviation of the process as calculated from the parent spectrum is σ x = 2.3365 . The standard deviations of the three simulated time series are 2.414, 2.328 and 2.3995. The discrepancy (less than about 3 %) is caused by the unduly coarse frequency segmentation.

A.4

The Cholesky decomposition

Given a positive definite and symmetric matrix X , the Cholesky decomposition of X is defined by a lower triangular matrix Y of the same size that satisfies the following:

X = YYT

(A.24)

Expanding this equation

 x 11    x i1   x  N1

x1i x ii x Ni

x 1 N   y11     x iN  =  yi1     x N N   y N 1

0 yii yN i

0   y11     0 ⋅ 0     y N N   0

y1 i yii 0

y1 N    yiN    y N N  (A.25)

and developing the matrix multiplication column by column, it is seen that the first column renders

 y11 = x 11  x 11 = y11 ⋅ y11   x = y ⋅y   21  y21 = x 21 / y11 21 11   ⇒    x N 1 = y N 1 ⋅ y11    y N 1 = x N 1 / y11

(A.26)

while the second column renders  y22 = x 22 − y21 y21  x 22 = y21 ⋅ y21 + y22 ⋅ y22    x = y ⋅y +y ⋅y   32  y32 = ( x 32 − y31 y21 ) / y22 31 21 32 22   ⇒    x N 2 = y N 1 ⋅ y21 + y N 2 ⋅ y22  y = (x − y y ) / y 22 N2 N 1 21  N2 and so on. This can be summarized as follows:

(A.27)

222


y11 = ( x 11 )

1/2

1/2

i −1   yii =  x ii − ∑ yik2  for i = 2,....., N − 1 k =1   j −1   yij =  x ij − ∑ yik y k j  / y jj for a ll i > j k =1  

yN N = ( x N N

)

1/2

(A.28)

Appendix B DETERMINATION OF THE JOINT ACCEPTANCE FUNCTION B.1

Closed form solutions

The calculation of wind load effects, static or dynamic, will inevitably involve the establishment of the joint acceptance function, normalised or non-normalised. As shown in chapter 2.10, it represents the statistical averaging in space, and it contains the integral

I (β ) =

1 1

∫ ∫ f ( xˆ ) ⋅ f ( xˆ ) ⋅ exp ( −β ⋅ ∆xˆ ) d xˆ d xˆ 1

2

1

(B.1)

2

0 0

where, f ( xˆ ) is some influence function or mode shape, xˆ is a non-dimensional coordinate between 0 and 1, ∆xˆ = xˆ1 − xˆ 2 and

C m n ω L exp / V if dyn a m ic r espon s x L exp / L m if st a t ic r espon s

m = u or w where  n = y f or z f

β =

(B.2)

Some closed form solutions (presented by Davenport [14], see also examples 6.1 and 6.2) are given below (and plotted in Figs. B.1 – B.3): Influence function or Mode shape, f ( xˆ )

Reduced integral, I ( β )

1

( 2 / β )  β − 1 + exp ( −β )

xˆ 2 xˆ − 1 sin ( n π xˆ )

B.2

2

( 2 / β ) ⋅  β / 3 − β / 2 + 1 − exp ( −β ) ⋅ ( β + 1 ) ( 8 / β ) ⋅  β /12 − β / 4 + 1 − exp ( −β ) ⋅ ( β / 4 + β + 1 ) 4

4

3

1

β 2 + (nπ )

3

2

2

2

2

2   2 (nπ )   ⋅ β + ⋅ − − ⋅ 1 exp cos β n π ( ) ( ) 2   β 2 + (nπ )  

Numerical solutions

In most cases a numerical integration is the most effective solution, in which case Eq. B.1 is to be replaced by:

224

B. DETERMINATION OF THE JOINT ACCEPTANCE FUNCTION

I (β ) =

1 N2

N

N

∑∑ f ( xˆ ) ⋅ f ( xˆ ) ⋅ exp ( − β ⋅ ∆xˆ ) p =1 k =1

p

k

(B.3)

where ∆xˆ = xˆ p − xˆ k and N is the number of integration points. It should be noted that in general a finely meshed integration scheme is required, i.e. a large N . The reason for this is of course that the exponential function is rapidly dropping at increasing values of its argument. The solution to a good number of cases has been plotted in Figs. B.1 – B.3:

Fig. B.1

Sinus type of typical mode shape functions

B.2 NUMERICAL SOLUTIONS

Fig. B.2

Cosine or polynomial type of typical mode shape functions

Fig. B.3

Linear type of typical static influence functions

225

Appendix C AERODYNAMIC DERIVATIVES FROM SECTION MODEL DECAYS From wind tunnel section model tests the aerodynamic derivatives were first quantified by the interpretation of in-wind simple decay recordings as described by Scanlan & Tomko [17]. From such testing six aerodynamic derivatives may be extracted, as shown in the following. The section model contains two intentional modes, one in the across wind vertical direction and one with respect to torsion, i.e.:

Φ ( x ) = [φ 1

φ φ2] =  z 0

0

(C.1)

φθ 

Internal unintentional flexibilities beyond those associated with these modes are most often insignificant, in which case φz ≈ φθ ≈ 1 . It is in the following taken for granted that their still-air properties

ω1 (V = 0 ) = ωz ω2 (V = 0 ) = ωθ

ζ 1 (V = 0 ) = ζ z

and

ζ 2 (V = 0 ) = ζ θ

  

(C.2)

are known, and that any additional response contributions from other modes are insignificant or have effectively been filtered off. The testing strategy is to set the section model into decaying free motion at a suitable choice of mean wind velocity settings. Idealised recordings from such a test are illustrated in Fig. C.1. The velocity dependent response curves may mathematically be fitted to r  r (V , x , t ) =  z  = Φ ( x ) ⋅ η (V , t ) (C.3) rθ  where:

cz   η  η (V , t ) =  1  = exp ( λr ⋅ t ) ⋅   η2  cθ ⋅ exp ( −i ⋅ψ r ) 

(C.4)

and λr (V ) = −ζ r ⋅ ωr + i ⋅ ωr , from which the in-wind damping ratio ζ r (V ) , resonance frequency ωr (V

)

and phase angle ψ r (V

)

may be quantified. The difference between

observed in-wind values of ζ r , ωr ψ r and their corresponding still-air counterparts will then contain all the effects of motion induced interaction between the section model and the flow. Since η (V , t ) has been idealised into a single harmonic component it is

necessary to assume that the motion induced part of the loading is dominant and narrow– banded, and that the buffeting contribution is insignificant or it has been filtered off. The

C. AERODYNAMIC DERIVATIVES FROM SECTION MODEL DECAYS

228

general equation of motion that contains all the relevant motion induced effects as expressed by the aerodynamic derivatives is then given by

⋅ η + C ⋅ η + K ⋅ η ≈ C ⋅ η + K ⋅η M ae ae  C ae  T  = ∫ Φ K ae  L exp

where

(C.5)

C  ⋅  ae  ⋅ Φ d x K ae 

(C.6)

Since the testing strategy only allows for the determination of six of the altogether eight motion induced load coefficients in the present set-up it is necessary to make a simplification. The following is adopted:

H C ae =  1  A1

H2  A 2 

and

0 K ae =  0

H3  A 3 

(C.7)

Fig. C.1 I.e., H 4

Typical decay recordings as obtained from section model tests; top diagram: vertical displacements; lower diagram: torsion and A 4 are discarded. Thus,

C ae zz C ae =  C ae  θz

C aezθ   φz2 H 1 = ∫  C aeθθ  L exp φθ φz A1

φz φθ H 2 

dx

φθ2 A 2 

(C.8)


229

and

0 = K ae 0 

K aezθ  0 φz φθ H 3  = ∫  dx 2 K aeθθ  L exp 0 φθ A 3 

(C.9)

The equation of motion is then given by:  M z   0

C z − C ae 0  zz  ⋅ + η   M θ   −C aeθ z

−C aezθ

  K  ⋅ η +  z   0 

Cθ − C aeθθ

− K zθ K θ − K aeθθ

 0   ⋅ η =   (C.10)  0 

Introducing K z = ωz2 M z , K θ = ωθ2 M θ , C z = 2 M z ωz ζ z , Cθ = 2 M θ ωθ ζ θ and that η = λr ⋅ η and η = λr2 ⋅ η , then the equation of motion is reduced into   C aezz  2ωz ζ z −  1 0  2  M z ⋅ + λ    r C aeθ z  0 1     − Mθ  

−

C aezθ M z

2ωθ ζ θ −

C aeθθ M θ

   ωz2    ⋅ λr +     0  

−

K aezθ

ωθ2 −

Mz K

aeθθ

Mθ

   ⋅η = 0     (C.11)

It is convenient to replace the aerodynamic load coefficients H j and A j , j = 1,2,3 , in Eq. C.7 with the non-dimensional quantities H *j and A *j called aerodynamic derivatives and defined by: H C ae =  1  A1 0 K ae =  0

 H 1* H 2  ρB 2 V ω = ⋅ ⋅  ( ) r A 2  2  B A1* 0 H 3  ρB 2 2 = ⋅ ωr (V ) ⋅   A3  2 0

B H 2*   B 2 A 2* 

B H 3*   B 2 A 3* 

(C.12)

(C.13)

Thus,

ρB 2 ⋅ ωr (V C ae = 2

 φz2 H 1* φz φθ B H 2*   dx * 2 2 * L exp  φθ φz B A1 φθ B A 2 

) ∫

and *

(C.14)


230

0 φz φθ B H 3   dx 2 2 * L exp  0 φθ B A 3 

(C.15)

j = z or θ

(C.16)

where

2  ∫ φz d x  2 L  β zz = ρ B ⋅ exp 2  mz ∫ φz d x  L   ∫ φz φθ d x  ρ B 3 L exp ⋅  β zθ = 2 m z  ∫ φz d x  L

(C.17)

where

 ∫ φθ φz d x  3 L exp ρ B  βθ z = ⋅ 2  m θ ∫ φθ d x  L  2  ∫ φθ d x 4  ρ B L exp ⋅  βθθ = 2 m θ  ∫ φθ d x  L

(C.18)

2 = ρ B ⋅ ω 2 (V K ae r 2

Defining

) ∫

m j = M j / ∫ φ j2 d x L

and the abbreviations

h 2 = β zθ h 3 = β zθ

a 3 = βθθ

2

ωr

 ⋅ A1*   ωr * ⋅ ⋅ A2  2   ωr2 * ⋅ ⋅ A3   2

a1 = βθ z ⋅ a 2 = βθθ

ωr

 ⋅ H 1*   ωr * ⋅ ⋅H2  2   ωr2 * ⋅ ⋅H3   2

h1 = β zz ⋅

2

then the equation of motion is given by

 1 0  2ω ζ − h1  λr2 +  z z   0 1  −a1  

ω 2 −h 2  λr +  z  2ωθ ζ θ − a 2   0

cz −h 3     0  =    2 ωθ − a 3   cθ ⋅ exp ( −iψ r )  0  (C.19)

Introducing exp ( −iψ r ) = cos ψ r − i ⋅ sin ψ r

{

}

 λ 2 + ( 2ω ζ − h ) λ + ω 2 c − ( h λ + h )( cos ψ − i sin ψ ) c  z z r z z r r θ 1 2 r 3  r  = 0   −a λ c + λ 2 + 2ω ζ − a λ + ω 2 − a cos ψ − i sin ψ c  0  ( θ θ 2) r θ 3 ( r r r) θ    1 r z

{

}

(

)

(C.20)

and that λr = ( −ζ r + i ) ⋅ ωr and λr2 = ζ r2 − 1 − i ⋅ 2ζ r ⋅ ωr2 , then the following is obtained:


 cz          

231

  2  h ω2 ω h h  ζ r − 1 + z2 − 2 z ζ z ζ r  + 1 cz ζ r + 2 cθ (ζ r cos ψ r − sin ψ r ) − 32 cθ cos ψ r  ωr ωr ωr ωr    ωr    2   ω  ω ω  cθ  ζ r2 − 1 + θ2 − 2 θ ζ θ ζ r  cos ψ r + 2 cθ  θ ζ θ − ζ r  sin ψ r +    ω ω ω   r r r    a a1 a  cz ζ r + 2 cθ (ζ r cos ψ r − sin ψ r ) − 32 cθ cos ψ r  ωr ωr ωr 

   ωz  h1 h h cz + 2 cθ (ζ r sin ψ r + cos ψ r ) − 32 cθ sin ψ r   −2 cz  ζ z − ζ r  + ω ω ω ωr  r  r r       0  2  =  −i ⋅  2   ωθ  ωθ ωθ  cθ  ζ r − 1 + 2 − 2 ζ θ ζ r  sin ψ r − 2cθ  ζ θ − ζ r  cos ψ r +      ω ω   0   ωr  r r      a1 c + a 2 c (ζ sin ψ + cos ψ ) − a 3 c sin ψ  r r r 2 θ  ωr z ωr θ r  ω   r (C.21) The tests comprise three different conditions of motion control. First the decay tests are carried out with the physical constraint that cθ = 0 . Under this testing condition the imaginary part of Eq. C.21 is reduced to

−2 cz (ζ z ωz ωr − ζ r ) + h1 cz ωr = 0 from which: and thus,

(C.22)

h1 = 2 (ωz ζ z − ωr ζ r )

(C.23)

 4  ωz  ζz − ζr  β zz  ωr 

(C.24)

H 1* =

The second series of decay tests are carried out with the physical constraint that cz = 0 , in which case Eq. C.21 is reduced to  2   ω  ω2 ω  ζ r − 1 + θ2 − 2 θ ζ θ ζ r  cos ψ r + 2  θ ζ θ − ζ r  sin ψ r +   ωr ωr    ωr      a 2 (ζ cos ψ − sin ψ ) − a 3 cos ψ  r r r  ωr r  0  ωr2     (C.25)  =    0    ωθ  ωθ2 ωθ  2    ζ ζ ζ ψ ζ ζ ψ 1 2 sin 2 cos − + − − − +     θ θ r r r r r    ωr ωr2  ωr       a2 a3   (ζ r sin ψ r + cos ψ r ) − 2 sin ψ r ωr  ωr 


232

a 2  2 (ωθ ζ θ − ωr ζ r )  a  =  2 2 2 r  3  ωθ − ωr − ωr ζ r 

Thus,

(C.26)

from which A 2* = A 3* =

 4  ωθ ζ − ζr   βθθ  ωr θ 

(C.27)

 2  ωθ2 2  2 − 1 − ζ r  βθθ  ωr 

(C.28)

After h1 , a 2 and a 3 have been determined then the third series of decay tests are carried out with no physical constraints, such that cz ≠ 0 and cθ ≠ 0 , in which case the full version of Eq. C.21 applies. Eliminating h 3 from the first real and imaginary parts and a 2 from the second real and imaginary parts then the following equations are obtained:           cθ 

      ωz  h1  h2  0  +cz 2  ζ z − ζ r  −  cos ψ r − cθ =  ωr  ωr    ωr     0    2    ωθ  a1 a ωθ2 ωθ ζ θ ζ r  + 2cθ ζ r  ζ θ − ζ r  + cz ζ r2 + 1 − 32 cθ   ζ r − 1 + 2 − 2  ωr ωr ωr   ωr  ωr   (C.29) 2 2 2 2 c ζ + 1 − ωθ ωr + a 3 ωr from which a1 = θ ⋅ ωr ⋅ r (C.30) cz ζ r2 + 1 sin ψ r   ω2 ω h cz  ζ r2 − 1 + z2 − 2 z ζ z ζ r + 1 ζ r  sin ψ r  ωr ωr  ωr 

(

(

h2 =

cz ωr cθ

)

)

 2   ω  h ωz2 ωz h ζ z ζ r + 1 ζ r  sin ψ r + 2  z ζ z − ζ r  − 1  ζ r − 1 + 2 − 2  ωr ωr  ωr   ωr  ωr 

   cos ψ r    (C.31)

Finally, h 3 may be determined from the first real part of Eq. C.21, rendering    h ω2 ω h ⋅  ζ r2 − 1 + z2 − 2 z ζ z ζ r + 1 ζ r  + 2 ⋅ (ζ r cos ψ r − sin ψ r )    ωr ωr  ωr ωr   (C.32) From Eqs. C.17 and C.18 h3 =

ωr2 cos ψ r

c ⋅ z  cθ

A1* =

2

⋅

a1

βθ z ωr

,

H 2* =

2

⋅

h2

β zθ ωr

and

H 3* =

2

⋅

h3

β zθ ωr2

(C.33)

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

[15] [16] [17] [18]

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Vickery, B.J. & Basu, R.I., Across–wind vibrations of structures of circular cross section. Part 2, Development of a mathematical model for full–scale application, Journal of Wind Engineering and Industrial Aerodynamics, Vol. 12 (1), 1983, pp. 79 – 97. Ruscheweyh, H., Dynamische windwirkung an bauwerken, Bauverlag GmbH, 1982, Wiesbaden und Berlin. Dyrbye, C. & Hansen, S.O., Calculation of joint acceptance function for line – like structures, Journal of Wind Engineering and Industrial Aerodynamics, Vol. 31, 1988, pp. 351 – 353. Selberg, A., Oscillation and aerodynamic stability of suspension bridges, Acta Polytechnica Scandinavica, Civil Engineering and Building Construction Series No. 13, Oslo, 1961. Shinozuka, M., Monte Carlo solution of structural dynamics, Computers and Structures, Vol. 2, 1972, pp. 855 – 874. Deodatis, G., Simulation of ergodic multivariate stochastic processes, Journal of Engineering Mechanics, ASCE, Vol. 122 No. 8, 1996, pp. 778 – 787. Hughes, T.J.R., The finite element method, Prentice-Hall, Inc., 1987. Scanlan, R.H., Roll of indicial functions in buffeting analysis of bridges, Journal of Structural Engineering, Vol.110 No. 7, 1984, pp. 1433 – 1446. Chen, W.F. & Atsuta, T., Theory of beam–columns, Volume 2, Space behaviour and design, McGraw–Hill Inc., 1977. Theodorsen, T., General theory of aerodynamic instability and the mechanism of flutter, NACA Report No. 496, Washington DC, 1934. Den Hartog, J.P., Mechanical vibrations, 4th ed., McGraw–Hill, New York, 1956. Cook, R.D., Malkus, D.S., Plesha, M.E. & Witt, R.J., Concepts and applications of finite element analysis, 4th ed., John Wiley & Sons Inc., 2002. Millikan, C.B., A critical discussion of turbulent flows in channels and circular tubes, Proceedings of the 5th International Congress of Applied Mechanics, Cambridge, MA, 1938, pp. 386 – 392.

INDEX Acceleration 44 Across wind direction 1, 92 111, 134135, 141, 147, 153, 197 Aerodynamic coupling 98 damping 106, 107, 124-125, 135, 148-150, 153-154 derivatives 97, 99, 100, 102,106-107, 112-113, 124, 135-136, 141, 195, 201, 205-207 mass 119, 129-130 Along wind component 58-59, 61, 64, 68, 125, 164, 189 direction 1, 6, 91-92 111, 117, 119, 143 Annual maxima 55, 56 Auto correlation 20 covariance 20, 22, 41, 60, 61 spectral density 33-35, 43, 210 spectrum 41 Averaging period 55 Axial force 182 component 70 Background part 79-80, 125, 146, 152, 159-160, 163, 166 response 166, 168 Bandwidth 152 Bending moment 10, 45, 50, 157, 164, 166, 169, 182 Bernoulli’s equation 1 Bessel 97, 100 Bluffness 106, 201 Bridge 1, 8, 55, 91, 98, 108-109, 111, 122-123, 134, 147, 153-154, 157, 167, 169-171, 179,-182, 190, 197, 205 Broad band process 30, 33, 57, 111, 125, 142

Buffeting 2, 91, 98, 112, 116-117, 127, 130, 132, 142, 182, 186 Cartesian 6, 8, 53 Centroid 9 Cholesky 214-216, 221 Coefficient 23-24, 47, 50-51, 82, 86-87 Coordinate system 6, 8, 10 Coherence 38, 43, 67, 106, 108, 144, 154 Correlation 19, 20, 23 Co–spectrum 40, 43, 67-68, 134 Coupling between components 76, 116 of modes 76, 82, 130, 140 Covariance 19, 39, 63, 130, 146, 166169, 171, 177, 178, 180, 182, 185, 187-190, 214 Critical velocity 196-197, 200 Cross correlation 23 covariance 23-24, 43, 63-65, 83, 130, 178 Cross sectional forces 7, 80,157 rotation 10, 111 stress resultant 2, 6, 9 Cross spectral density 38, 83 spectrum 41, 66-67, 118,132-134, 140, 142, 144 Cumulative probability 13, 55 Cut-off frequency 211 Damping coefficient 106-107 matrix 75-76 properties 4, 98, 105 ratio 4, 75, 77, 106, 198 Decay curve 66 recording 98, 227-228 Den Hartog criterion 201

236 Design period 157 Displacement components 70, 73, 76, 81, 83, 102, 162 response 5, 76-77, 79-81, 161, 182 Divergence 112, 197, 199-200 Drag coefficient 91, 134 component 8, 160 force 92, 97, 109 load 76 Dynamic amplification 11 response 1-2, 69, 76, 78, 109, 119, 125-127, 134-135, 141-142, 146, 152-156 Eigen damping 4, 105, 154, 156, 198 frequency 3, 70, 75-77, 81, 98, 103104, 116-117, 122, 134, 147, 153, 197-205 mode 3, 70, 75-76, 159-160 value 3, 70-73, 75, 90, 196-197 vector 72-73, 75 Element forces 162-163, 177 Ensemble statistics 6, 15 Equivalent mass 107, 119, 124, 135, 147, 154, 186, 190 Ergodic process 30, 44 Euler constant 32, 57 formulae 36 Extreme value 10-11, 27, 30, 57-58 weather condition 54, 56 Failure 157 Finite element 75, 113, 160-163, 174 Fisher-Tippet 56 Flow axes 7, 92 component 58, 93, 97, 128 direction 1, 6-8, 53-55, 61, 65, 91 exposed 117, 123, 134, 144, 154, 165 incidence 91, 93 Fluctuating

INDEX

components 1, 11, 70, 93, 118, 179, 186, displacement 5, 103 force 1, 97, 182 load 2, 103-104, 127, 160, 165-166 part 1, 2, 5, 6, 8, 10-11, 58, 69, 9195, 111,116, 157-161, 163 wind velocity 5, 174 Flutter 197, 199, 203-206, 208 Force components 114, 157, 160, 162, 166, 173, 177-178, 182-188 Fourier amplitude 36, 39 43, 78-79, 83, 88, 117, 128 component 36-37 constant 38 decomposition 35 transform 34, 43, 78, 82, 85, 96, 117, 128, 132, 142-143, 184 Frequency domain 2,70, 73, 78, 80, 83-85, 9698, 105, 110-111, 116-117, 130, 146, 152 158-160, 185 segment 210-211, 215 Frequency response function 79, 112, 116-117, 119-120, 125-126, 129, 137, 140 matrix 87-88, 130, 142 Full scale 62, 66 Galloping 197, 199-202 Gaussian 5, 11, 14, 29, 110, 157 Global 161-163 Harmonic component 34-35, 39, 210, 228 Hermitian 214 Homogeneous 5, 15, 53, 60, 63, 91, 110, 157 Horizontal element 114, 174, 176 Identity matrix 73, 87, 130 Imaginary 36, 40, 43, 67, 78, 201-202, 206-207 Impedance 195, 197, 199-200, 202 Influence function 46, 160, 164-166, 170-174

INDEX

Instability 100, 112-113, 195, 197, 199205 Instantaneous velocity pressure 58, 91 wind velocity 1, 174 Integral length scale 62, 65-66, 114, 123, 135, 144, 152, 173 Isotropy 60 Joint acceptance function 47-48, 118, 120122, 124-126, 129, 133, 138, 170, 172-174, 193, 223 probability 13 Kaimal spectrum 62 Length scale 61, 63, 65, 97 Lift 76, 91, 106, 110 coefficient 91, 106 force 92, 97 load 76, 110 Linear 1, 11, 90, 93, 157-158 Linearity 83, 79 Line like structure 1, 8, 70, 103, 109, 157, 163, 182 Load 69, 73 coefficient 86-87, 91, 93-94, 97, 134, 167 component 70, 73, 113-115, 160, 162, 175-176, 180 vector 75-76, 85, 87, 113, 115-116, 143, 161, 175, 177, 179 Lock–in 104, 106, 142, 154 Long term statistics 4, 6, 58 Main structural axes 8-9 Mean load 113-114 value 1, 13, 70, 93, 111, 113, 157158, 163 wind velocity 1, 53-57, 61, 98-100, 112, 125-126, 134, 140-141, 154-156 Modal displacement 90, 182 damping 75, 106 load 75, 79, 87-88, 132

237 mass 72, 75, 98 stiffness 75 Mode shape 70, 72-76, 78, 81, 83, 90, 99, 117, 123, 134, 140-141, 143, 145, 147, 151, 153, 183, 188, 191 Modulus 40 Moment coefficient 91,106 force 92, 97 load 76, 110, 160 Motion induced 2, 76-78, 81-82, 84-85, 87, 112-113, 116-117, 140-142, 195, 199 Multi mode 76, 84, 87, 130, 132, 183 Narrow band process 29, 146, 152, 159, 196 Neutral axes 8 Non-coherent time series 213 Orthogonal component 1, 70, 75 Parent population 55 variable 35 Peak distribution 31 factor 11, 33, 58, 111, 142, 158 value 27 Phase 40, 78, 85 angle 211, 215 spectrum 40, 67 Probability density function 13 distribution 13, 16, 19, 26, 56, 59 Quad spectrum 40 Quasi static 79, 98, 102, 124-125, 135, 141, 159-160, 163-164, 177, 201, 203 Random variable 4, 13-14, 19-20 Rayleigh 14, 19, 29, 55, 75 Reference height 55 point 59 Representative condition 54 Resonance

238 frequency 98-99, 100, 112-113, 134, 140-141, 196, 200-202 velocity 154-155 Resonant part 79-80, 125, 146, 152, 159-160, 182, 184-185 Response calculation 69, 73, 76, 79, 81, 84, 8788, 109-113, 117, 122, 127, 130, 142, 151 covariance 76, 83, 130 matrix 130, 139, 142, 145-146 spectrum 111, 117, 123, 125-126, 140, 142, 151 Return period 57 Reynolds number 107-108 Root coherence 43 Roughness length 55 Safety 11, 53, 157 Sampling frequency 58 Sears function 97 Section model 97-98, 227-228 Separation 25, 63, 86, 118, 121, 166, 168, 179, 181 Selberg 205 Shear centre 8-9, 69, 75 Short term 4, 15, 18, 21, 23, 30-31, 45, 110 Simulation of random process 209 Single degree of freedom 76, 83, 117 point statistics 58 point time series 210 point spectrum 67, 125-126, 137, 144 Spatial averaging 45, 134, 165-166 properties 63, 65 separation 48, 118, 179 Spectral decomposition 210 density 33-36, 38-39, 43-44, 62, 89, 125, 134, 137, 139, 184-185, 214 moment 45 Spectrum double-sided 37, 39 single-sided 34, 40 Stability limit 112, 196-197, 201-205

INDEX

Standard deviation 14, 24, 111, 119, 123, 125, 127, 129, 141, 155, 157, 159 Static response 109, 112 stability 196 Stationary 2, 5, 11, 15, 44, 53, 63, 91, 110, 157 Stochastic process 2, 4, 11, 15, 18 variable 30, 33, 41, 45-46 Stress resultant 7, 9, 10, 157, 182-183, 185 Strouhal number 103, 108 Structural axis 7, 91, 93, 97 damping 75 displacements 1, 7, 73, 78, 85, 91, 93, 104-105, 109, 142, 158 mass 78 stiffness 112, 158 strength 10 Taylor 61, 65 Theodorsen 100, 205 Threshold crossing 27-28 Time domain 2, 6, 15, 62, 77, 85, 110-111, 157-159, 165, 209 lag 20, 25, 60, 63 scale 61 step 211 Torsion 202 mode 77 moment 10, 103, 106, 157, 166, 169, 183 response 140, 150 stiffness 71 Total load 76-77, 81 response 76, 84, 111, 159 Tower 1, 8 Turbulence component 1, 53, 58-62, 64, 66, 68 intensity 59, 118, 123, 135, 166 length scale 61

INDEX

profile 54

239 Vortex shedding 2, 102-108, 111-112, 142-143, 146-147, 150, 152-156, 182

Unstable behaviour 98, 112 Variance 13-14, 16-19, 33-34, 36-38, 46, 48, 111, 117-118, 130, 146, 148, 151, 165-169, 182, 185, 188-189 Velocity pressure 1, 56-58, 91, 112 profile 54-55 vector 1, 6, 8, 53, 91-92, 174 Vertical element 114, 174, 176 Viscosity of air 108 Von Karman spectrum 62-63

Weibull 14, 19, 55 Wind climate 54 direction 108 force 2, 103 load 45, 91, 102 load component 176 profile 54-55 velocity 1, 2, 15, 18, 53, 92, 98-100, 104-105, 112-113, 125, 127 Wind tunnel 97-98, 141, 154 Zero up-crossing 29, 45

Theory of Bridge Aerodynamics

Theory of Bridge Aerodynamics

Aerodynamics

Aerodynamics: Fundamentals of Theory, Aerodynamics of an Airfoil and Wing; Methods of Aerodynamic Calculation

Aerodynamics of the Airplane

Aerodynamics of the Aeroplane

Fundamentals of Aerodynamics

Aerodynamics of the Aeroplane

Transonic aerodynamics: problems in asymptotic theory

Bridge

Theoretical aerodynamics

Building Aerodynamics

Transonic aerodynamics

Bridge

Bridge

Bridge

Bridge

Bridge

Bridge

Bridge

Bridge

Principles of ideal-fluid aerodynamics

Aerodynamics of Wings and Bodies

Fundamentals of Aerodynamics,Third Edition

Fundamentals of Modern Unsteady Aerodynamics

Introduction to Aerodynamics of Flight

Fundamentals of Modern Unsteady Aerodynamics

Principles of Ideal-Fluid Aerodynamics

Principles of ideal-fluid aerodynamics

Early Developments of Modern Aerodynamics

Bridge of Souls

Theory of Bridge Aerodynamics