ELSEVIER Journal of Wind Engineering

and Industrial Aerodynamics 57 (1995) 281-294

dOURN~_ OF

A generalized model for assessment of vortex-induced vibrations of flexible structures

Allan La r s en

COWlconsult Consulting Engineers and Planners A/S, Parallelvej 15, 2800 Lyngby, Denmark

Abstract

The present paper proposes a one degree of freedom (1DOF) non-linear model of self limiting cyclic wind loads for application in finite element method analyses of light structures subjected to vortex shedding excitation under lock-in conditions. Being empirical by nature, the model includes three independent parameters to be determined from response tests with representative aero-elastic wind tunnel models or prototypes. The paper discusses methods for parameter identification from response data. The paper also evaluates the proposed load model versus other 1DOF empirical vortex shedding models which have found some acceptance in wind engineering.

Notation

a(t)

ao e(t)

f g

t

CL Cs = Sc = 4n~sM/pD 2 Ca = 4 n ( a M / p D 2

D

FG,V,F M ~x

P r/

"theoretical" envelope initial value, "theoretical" envelope "experimental" envelope frequency (Hz) peak factor time lift coefficient Scruton number aerodynamic Scruton number crosswind body dimension crosswind vortex shedding force generalized mas/uni t length rms limiting amplitude asymptotic value, "theoretical" envelope aerodynamic non-linearity parameter air density non-dimensional crosswind response

0167-6105/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0167-6105(95)00008-9

282 A. Larsen/J, Wind Eng. Ind. Aeroc(vn. 57 (1995) 281 294

~0

p = pDZ/M

(I)

phase aerodynamic exponent mass ratio structural damping relative to critical aerodynamical damping relative to critical circular frequency (rad/s) functional relationship

1. Introduction

Flexible and lightly damped slender structures such as bridges, chimneys and towers are often found to be prone to vortex-induced vibrations at relatively low wind speeds. The bluff shape of most practical cross sections promotes formation of periodic and coherent "vortex streets" in the wake of the structure. Large amplitude resonant vibrations may occur at wind speeds where the vortex shedding action locks on to one of the crosswind vibration modes of the structure. At this condition commonly referred to as lock-in, vortex-induced vibrations are found to be self- limiting, amplitude dependent and highly sensitive to the mass density and the inherent damping level of the structure.

The finite element method (FEM) has received broad acceptance as the foremost analysis tool in contemporary structural engineering. FEM analyses allow detailed computations of the overall response and stress distributions in critical structural members subjected to deterministic or random loads. Hence FEM computations appear as the logical choice for assessment of vortex shedding action on slender structures and for evaluation of alternative damping measures intended for sup- pression of excessive vortex induced responses.

The objective of the present paper is to establish a suitable forcing function model which recognizes the non-linear amplitude dependent character of vortex shedding action. Furthermore, the model must be capable of reproducing the functional relation- ship between response and mass density/structural damping as established through wind tunnel tests of a given structure. Finally, the proposed model must comply with the load generation facilities commonly available in commercial FEM codes.

2. I D O F vortex shedding models in the literature

Structural loads are usually applied in dynamic FEM analyses as external time dependent forcing functions of a predefined magnitude and are thus independent of the structural response. Scruton [1] discussed the application of this approach (the simple forced lift oscillator at resonance) for mathematical modelling of crosswind forces due to vortex shedding. He concluded that the resulting hyperbolic relationship between vortex induced response and structural damping was not consistent with wind tunnel observations of cylinders oscillating at finite amplitudes. The resonant forced lift oscillator (presently referred to as FLO model) was later applied by Smith

A. Larsen/J. Wind Eng. Ind. Aerodyn. 57 (1995) 281-294 283

and Wyatt [2] as a convenient framework for correlation of wind tunnel test results obtained in connection with drafting of the proposed British design rules for bridge aerodynamics. The hyperbolic relationship between bridge response and structural damping is thus reflected in empirical formulae given in the above mentioned design rules.

Amplitude dependent loads may be introduced in FEM models as negative damping elements (dashpots) provided the dynamic loads can be expressed in terms of the local vibration velocity of the structure. In a review of vortex induced flow phenomena, Marris [3] proposed to express vortex induced crosswind forces at lock-in as a cubic function of the structural vibration velocity. For simple harmonic motion the cubic model may be rearranged to yield a linear forcing term combined with a non-linear amplitude dependent restoring term known from the classical Van der Pol oscillator. The latter form was proposed by Simiu and Scanlan [4] for projection of bridge section model test results to prototype responses. This correction procedure, used in Ref. [4], involves allowances for differences in mode shape, mass density and damping between (section)-model and prototype. The Van der Pol oscillator concept (presently referred to as VPO model) was later utilized by Vickery and Basu [5], but within the framework of stochastic vibration theory, for assessment of vortex induced crosswind vibrations of chimneys and towers mainly of circular cross section.

3. A generalized Van der Pol model for vortex induced forces

The functional relationship between response and structural damping at lock-in is not satisfactorily accounted for by the resonant forced lift oscillator (FLO) and the Van der Pol oscillator (VPO) models as will be demonstrated in Section 6. Hence a generalization of the Van der Pol model for the crosswind force F~ due to vortex shedding action is proposed which allows improved adaption to experi- ments,

FG = #fCa(1 -- el~/12v)0, (1)

where r/, 0 are the structural displacement and velocity non-dimensionalized by the crosswind dimension of the structure. # = pD2/M,fare the mass ratio and oscillation frequency, respectively, and Ca, e, v are non-dimensional aerodynamic parameters to be determined from measurements of structural response under representative wind conditions. In FEM applications an alternative expression for FG is useful,

Fc = I~fCa 1 -- (2v + 1 ) ~ f ) 2~) q" (2)

Expressions (1) and (2) are equivalent provided the vortex induced response is harmonic with a well-defined frequency f, which often is the case in practical applica- tions. The non-linear forcing function given by (2) is readily modelled by non-linear dashpot elements available in a number of commercial FEM codes.

284 A. Larsen/J. Wind Eng. Ind. Aerodyn. 57 (1995) 281 294

4. Structural response versus structural damping

The functional relationship between steady state structural response and structural (viscous) damping predicted by the proposed GVPO model is obtained by equating the energy supplied by the vortex shedding process to the energy dissipated by the non-linear restoring force and structural damping over one period of oscillation as outlined in Appendix A.

Assuming that the vortex induced response behaves sinusoidally in time, the following expression is obtained for the steady state resonant amplitude,

Sc~] x':Y t3)

where the integral lc(v) = ~2. sin2(p)i cos(p)12v dp in general must be evaluated numer- ically. Sc = 4rr M~s/pD 2 is the Scruton number as proposed by Zdravkovich [6]. Eq. (3) is identical to the steady state response equation presented by Simiu and Scanlan (see Ref. [4], p. 205), for the case v = 1.

Fig. 1 displays normalized steady state response amplitudes rl/qL as function of the ratio of structural damping to aerodynamic damping SC/Ca, for representative values of the power multiplier v = 0.25, 0.5, 1.0 and 1.5. It can be noted that v governs the curvature of the response curves. Sc/Q = 1 defines the value of the Scruton number necessary to eliminate vortex induced responses. Sc = 0 defines the limiting amplitude qL = [rr/Ic(v)~] 1/2~ used for normalization of the response, q/qL = 1 would be sus- tained for the theoretical case of vanishing structural damping or structural mass.

The presence of the trigonometric integral Ic(v) in (3) which requires a priori knowledge of the power multiplier v appears slightly awkward, z/Ic(v) may however be approximated by the linear expression 3v + 1. This approximation is accurate

q

rl L \

1) 5 -

V

() f)2

1 I

\ \ \

0.25

I

~4

- - . - - . v 1.5

\ \ -- v - 1.0, ,

" ..... v 0.5 \ ~

()(~ ().8

Sc/C

Fig. 1. Normalized steady state response for the GVPO model for selected values of the power multiplier.

A. Larsen/J. Wind Eng. Ind. Aerodyn. 57 (1995) 281-294 285

within 8% in the interval 0 < v < 1 spanning the range likely to be encountered in practical applications.

5. Parameter identification, steady state response tests

The proposed empirical model for vortex shedding excitation includes three inde- pendent aerodynamic parameters Ca, e, v to be determined from physical experiments. Eq. (3) suggests that this task may be accomplished by measuring the steady state vortex induced response of a representative section model or aero-elastic model of the structure at a minimum of three different Scruton numbers. The aerodynamic para- meters are then identified by matching the measured response to the response equation (3).

Larsen [7] suggested a parameter identification method based on successive substi- tution of measured response and damping values in (3). This method assures a perfect fit of the response equation to three independent sets of Scruton numbers and measured responses (Sci, ~h), i = 1,2,3. If more than three data sets are available it becomes necessary to select three representative sets and thus attach lesser signifi- cance to the remaining data. Furthermore, experience has shown that the proposed identification method may yield difficulties for the estimation of v if certain ill conditioned data sets are applied.

Based on the experiences quoted above it was deemed appropriate to apply a least squares method for matching of experimental data to the GVPO response equation (3). The least squares fit is effectuated through an error expression comprising the sum of squares of differences between the measured responses ~h(Scl) and predicted responses r/(Sci, Ca, e, v) according to (3),

SSQ = ~ [rh(Sc,) - ~/(Sc,, Ca,e,v)] 2. (4)

The solution proceeds by establishing the values of Ca, e, v for which the sum of squares (4) is a minimum, a task accomplished by standard data reduction routines.

6. Evaluation of the GVPO model versus experimental data

Evaluation of the proposed GVPO model versus experimental data and the VPO and FLO models mentioned in Sections 3 and 4, was carried out by applying the analytical response models to data obtained at the National Maritime Institute by Walshe [8] for a deep trapezoidal bridge girder (model 5, Cleddau Bridge). The characteristics of the different models become immediately obvious when comparing predicted response curves in Figs. 2 and 3 to the experimental data (denoted by × ).

From Figs. 2 and 3 it can be noted that the predicted GVPO response curve (full curve) maintains a slight downward curvature and appears well adopted to the trend suggested by the experimental data. Response predictions following the VPO model (dashed curve) display an upward curvature which is not quite consistent with the experimental trends. Furthermore, the predicted Scruton number for zero response is

286 A. Larsen/J. Wind Eng. Ind. Aerodyn. 57 (1995) 281-294

0 . 1 ~ I I I I I

() l() 20 ~11 40 50 6(1

Sc

Fig. 2. Best fit of GVPO (solid curve), VPO (dashed curvei and FLO (dotted curve) models to measured data ( × ) for a trapezoidal bridge section. Smooth flow.

().15

(LI

1) ()

I I ~ . , I

I() 2f l ~() 4()

SC

Fig. 3. Best fit of GVPO (solid curve), VPO (dashed curve) and FLO (dotted curve) models to measured data ( x ) for a trapezoidal bridge section. Turbulent flow.

lower than the zero response Scruton number established during measurements a fact to be kept in mind in case the VPO model is applied in studies of damping

devices for suppression of vortex shedding excitation of structures. The FLO model (dotted curve) maintains a downward curvature by virtue of its hyperbolic shape, but fails to identify a Scruton number for vanishing vortex induced response.

Experimental response data for a straight circular cylinder and a tapered circular cylinder oscillating crosswind and perpendicular to their spanwise axes are reported by Scruton [1 ] . Best fits to the GVPO response equation and to the hyperbolic FLO

A. Larsen/J. Wind Eng. lnd, Aerodyn. 57 (1995) 281-294 287

0.4

0.3

0.2

O. 1

I I I

; . . . .

0 lO 20 30 40 50 60

Sc

Fig. 4. Best fit of GVOP (solid curve) and FLO (dotted curve) models to measured response data ( × ) for a straight circular cylinder.

11

0,4

0.3

0.2

O. 1

' I I I

1

10 20 30 4(}

Sc

Fig. 5. Best fit of GVOP (solid curve) and FLO (dotted curve) models to measured response data ( x ) for a tapered circular cylinder.

model are presented in Figs. 4 and 5 for further comparison. It can be noted that the G V P O model is well adapted to the measured response curves for the straight cylinder as well as for the tapered cylinder. The F L O model only resembles the data measured for the straight cylinder.

7. Parameter identification, transient response tests

Estimation of G V P O parameters f rom steady state response tests requires a min- imum of three individual experiments conducted at three different values of the

288 A. Larsen/J. Wind Eng. Ind. Aerodvn. 57 (1995) 281 294

Scruton number. Additional data sets are, however, often desirable in order to enhance the confidence level, which, in turn, leads to expansion of the wind tunnel test programme. Reduction of the experimental programme to comprise only one mass/damping condition is possible if estimation of the GVPO parameters relies on transient response measurements at lock-in.

Application of the Krylov Bogoliubov technique [9] to the aerodynamic and structural damping forces allows the derivation of an analytical expression for the envelope a(t) of the resonant transient response of the generalized Van der Pol oscillator. Following the procedure outlined in Appendix B, a(t) is obtained as

a(t) - [1 [1-[/4/ao)Z"]exp(. v l"PD2"~lC, Sc),) J 1/2'' (5)

where a0,/~ are the magnitudes at t = 0, t --, 1_ andf is the frequency of oscillation (in Hz). It can be observed that the envelope expression (5) includes the three aero- dynamic parameters Ca, e, v needed for steady state response predictions. Here g = /r(] - - S c / C a ) / | c ( v ) f l 2v by virtue of (3). Parameter identification may thus be accomplished by a least squares fit of (5) to experimental envelopes derived from measured "decay-to-resonance" or "growth-to-resonance" displacement traces. As an example of the three elements involved in the identification procedure, Fig. 6 shows a plot of the "theoretical" envelope a(t) according to (5), superimposed on a simulated "decay-to-resonance" trace q(t) and the derived "experimental" envelope e(t).

Excellent agreement is noted between the "theoretical" envelope and simulated decay trace. The "experimental" envelope curve e(t) is tangent to the peaks of the simulated decay trace as expected, except at the very start and end. Here the "experimental" envelope becomes "noisy", most likely due to the dual Fourier

().

T1

O.05

-0.05

- I ) 1 I) 2 4 ,~ X ll) 12 14 1(~ IN 20

t

Fig. 6. Simulated decay trace ~/(t) (solid curve), "experimental" envelope e(t) (dotted curve) and "theoret- ical" envelope a(t) (dashed curve) for the GVP O model.

A. Larsen/J. Wind Eng. Ind. Aerodyn. 57 (1995) 281-294 289 transform operation involved in the processing of the decay trace. The "noisy" parts should be discarded before carrying out the least squares fitting operation.

The simulated decay trace shown in Fig. 6 was obtained from a Runge-Kutta solution of a second order differential equation including the GVPO forcing func- tion (see Appendix B, Eq. (B.1)). The aerodynamic parameters Ca, e, v were derived from the smooth flow bridge section data given in Fig. 2. Mass, damping and stiffness parameters in the governing differential equation were chosen to yield a natural frequency f = 4 Hz and a Scruton number Sc = 15. The "experimental" envelope e(t) was derived from the simulated decay trace rift) using the Hilbert transform technique.

8. Vortex induced vibration of steel chimneys

Full-scale response data for vortex induced vibrations of slender structures cover- ing a representative range of Scruton numbers are rare. A notable exception is the data base compiled by Pritchard and elaborated by Daly [10] for 64 steel chimneys of circular cross section. Daly applied a least squares analysis to the Pritchard data in order to calibrate the Vickery-Basu response model [5] to aerodynamic conditions prevailing at supercritical and transcritical Reynolds numbers. Daly's analysis forms the basis of the CICIND design approach for evaluation of the crosswind response of prototype steel chimneys.

The Vickery-Basu model yields a response equation of the following form,

CL (6) r/= g ~/Sc - Cal-1 - (r//g~)2-I '

where g = x/2 { 1 + 1.2 arctan [0.75(SC/Ca) 4] } is a peak factor ranging between 1.41 and 4, CL is a lift coefficient accounting for forced background excitation and a--rlL/g, the rms value of the limiting amplitude mentioned in Section 4. The self-limiting Van der Pol term comprising Ca and r/2 is recognized in the denominator of (6). Solving for ~/yields the following fitting function,

( (Sc_ Ca)(~2 )1/2 r/ = g 2 C a + ~ / E - ( S c - Ca)O~2/2Ca] 2 + C 2 ~ 2 / C a . (7)

Eq. (7) is used in a least squares analysis, in conjunction with relevant prototype data, for determination of the full-scale ~, Ca, CL values sought after.

Introduction of the power multiplier v in (6) yields an equivalent response equation adopted to the generalized van der Pol model introduced in Section 3. This response equation cannot, however, be solved explicitly for ~/to yield a fitting function similar to (7). In order to proceed it becomes necessary to assume that the forced excitation at high Scruton numbers is negligible, i.e. CL ~ 0. The following fitting function is thus obtained,

r /= g~t(1 - Sc/Ca) 1/2~. (8)

290 A. Larsen/J. Wind Eng. Ind. Aeroclvn. 57 (1995) 281-294

q

0.6

().4

0.2

I I I I

\ ~ \× t__._.o.6 3

' ~ ~ Re x 10 6

() q l() I "~ 2(1 2s

Sc Fig. 7. Best fits of GVPO response model (solid curve} and Vickery Basu response model (dashed curve) to prototype chimney data. Supercritical flow 6 x l0 s < Re < 3 z 106.

q

O.2 U ~ \

t

11 o

I I I

I

'~ \ \ , I

* ~ .. ~ Re x ~o 6 ; : ~

I I m

q 111 IS 20 25

Sc

Fig. 8. Best fits of GVPO response model (solid curve} and Vickery Basu response model (dashed curve} to prototype chimney data. Transcritical flow Re > 3 x 106

For the purpose of comparison of the VPO and GVPO model concepts, limiting rms amplitudes ~ = 0.4 and ~ = 0.2 have been applied in the present analysis of supercriti- cal (6 x t0 5 < Re < 3 x 10 6) and transcritical (Re > 3 x 10 6) flow conditions. This choice is in accordance with the considerations given by Daly. Figs. 7 and 8 show plots of the Vickery Basu response curves (VPO model) (dashed curves) and response curves based on the GVPO model (solid curves) superimposed to the Pritchard data (denoted by x ). It can be noted that the Vickery-Basu model displays a very steep "transition region" between the response governed by self-limiting motion (low Scruton numbers) and forced motion (high Scruton numbers). In contrast, the GVPO

A. Larsen/J. Wind Eng. Ind. Aerodyn. 57 (1995) 281-294 291

response curves display a smooth transition more in line with experimental observa- tions for single cylinders shown in Figs. 4 and 5.

9. Summary and conclusion

The present paper has reviewed two 1DOF mathematical models for vortex shedding response available in the literature. Evaluation of these models versus available response data reveals that the functional relationship between the Scruton number and response amplitudes does not match the experimental trends in a satisfac- tory way.

A new 1DOF empirical model is suggested for vortex shedding excitation of light and flexible structures. The model is based on a generalization of the well-known Van der Pol type forcing function valid for self-excited and self-limiting vibrations. The mathematical form of the forcing function is well suited for design of non-linear dashpot elements applicable to FEM models of wind sensitive structures. Evaluation of the proposed model versus experimental data obtained from testing of wind tunnel models and prototype structures demonstrates that the curvature of the predicted structural response versus the Scruton number may be well adapted to experimental observations. This finding suggests that the proposed model is suitable for theoretical studies of damping measures for suppression of vortex induced vibrations of light flexible structures. Fig. 9 summarizes models for crosswind vortex shedding forces, predicted shapes of responses versus Scruton number and the number of aerodynamic parameters to be identified from wind tunnel tests.

Force/unit length 1 DOF response Parameters Proposal

F F = Kasin(tot ) + (/zfC'a';?)

F v = ~ f C a ~ _ , ~ 2 ] ~

FG = /L fCa[1-

"r/ , FLO i

° ' . . . . . SC

,r/ , VPO

\ \ Sc

" r / ".-, GVPO

Ka, (C'a)

O a l

C a j ,~ p I,I

Scruton 1963 Scanlan 1976 Codes

Marris 1964 Scanlan 1976 Vickery/ Basu 1983

Present

Fig. 9. Summary of 1DOF vortex shedding response models considered in the present paper.

292 A. Larsen/J. Wind Eng. lnd. Aerodyn. 57 (1995) 281 294

Acknowledgement

The author would like to take this opportunity to express his sincere thanks to Jannie Ohlendorff for pointing out a number of idiosyncrazies in his use of the English language and to Lone Holst-Jensen for excellent draftmanship.

Appendix A: Steady state response

The steady state response amplitude of a 1DOF oscillator subjected to the G V P O forcing function is obtained from the equation of motion,

FI + Ia.lC~O + (2~,1)2~/= I~.IC,(I - cl~/t2"j#. (A.I)

For harmonic resonant motion *1 = qo cos (2=Jt) the inertia and stiffness terms cancel out leaving the response to be governed by damping (0) terms,

/*f[(Sc - Ca)# + eCalr/12v#] = 0. (A.2)

Assuming that I q 12" can be approximated by lmqg ~', i.e. a weighting factor times the steady-state amplitude raised to the power 2v (A.2) yields

(+ ),: r/o = (1 - Sc/Ca) . (A.3)

The weighting factor lm is obtained by equating the energy content of the non-linear term [ql2v0 during one cycle to the energy content of the approximation Imr/2v0, i.e.

2r~ 2r~

2v 2 J 2v 2 i lmtlO Iio sinZ(p)dp = Icos qo rio (p)le"sinZ{p)dp,

0 ~

2~

f lcospl2"s in2(pldp

lm 0 IC{V) = 2~ - (A.4) 7r

f sin (p)dp

0

Hence the steady state response equation becomes

r/0 = (1 - Sc/ 'Q) , (A.5)

which is identical to (3). The trigonometric integral Iclv) must in general be evaluated by numerical

methods.

A. Larsen/J. Wind Eng. Ind. Aerodyn. 57 (1995) 281 294 293

For v = 1 (the V P O model) an analytical evaluat ion of Ic(v) is possible, i.e.

2n

f 1 Ic(v) = cosZ (p)sinZ(p)dp = -~ n,

0

leading to

1/o = x/(4/D(1 - Sc/Ca)

as given by Simiu and Scanlan [4].

(A.6)

Appendix B: Transient response

The Kry lov -Bogo l iubov technique [9] is applicable to weakly non-l inear second order differential equat ions of the type

for which an almost harmonic solution r/(t) = a(t)cos[o~t + q)(t)] is postulated. The ampli tude a(t) and phase to(t) are slowly varying functions of time which are assumed to change very little over one period 2n/o) of oscillation.

The first order differential equat ions for determinat ion of a(t) and t0(t) are as follows,

da ~c - 2cofl(a), (B.2) dt

dto x d~- = - 2--~ g l (a), (a.3)

where f l (a), g l (a) are coefficients of the Four ier series expansion for o~,

2~

1 f s i n p o ~ ( a c o s p , ae~sinp)dp, (B.4) A(a) 0

2n

1 f cos p o~ (a cos p, _ ae)sinp)dp. (B.5) gx(a) =-~

0

Now taking

~ ( r / , O) -- [(Ca - Sc)O - Ca~ir/12"O] (B.6)

yields

Ic(v)~C. 2~'~ f l (a) = aeo -- (C a - - S C ) + ~ a ) ,

gl(a) = O.

294 A. Larsen/J. Wind Eng. Ind. Aerodyn. 57 (1995) 281-294

I n t r o d u c t i o n o f f l ( a ) in (B.2) yields

dt - ~Ka (Ca - Sc) - ~:Ca a 2~" , (B.7)

tha t with the aid of (A.5), (B.7) can be r ea r r anged to read

da ~C(Ca-- SC) 2v d t - ~ f l ~ a ( a _ f12,), (B.8)

where fl is the m a g n i t u d e of the t r ans i en t r esponse for t ~ 0o. Eq. (B.8) c an be

in t eg ra t ed after s e p a r a t i o n of the var iables ,

da _ (~c(Ca Sc}dt f a(a2~ _ f l zv ) J -2flJ~v " (B.9)

The in tegra l in the le f t -hand side of (B.9) is o b t a i n e d f rom s t a n d a r d m a t h e m a t i c a l tables, l ead ing to

In [1 - ( f l /a) 2v] = - ~¢(C a Sc) t + cons t . (B.10)

I n t r o d u c i n g a = ao for t = 0, the t ime d e p e n d e n t enve lope a ( t ) is f inal ly o b t a i n e d as

a ( t ) = f l / { l - [1 - ( f l / a o ) Z " ] e x p [ - v f # ( C , - Sc) t ]} 1/2v (B.11)

r e m e m b e r i n g tha t ~" =/~1~ (B.1 1) is ident ica l to (5).

References

[1] C. Scruton, On the wind-excited oscillations of stacks, towers and masts, in: Wind effects on buildings and structures, Vol. I (National Physical Laboratory, Teddington, UK, 1963).

[-2] B.W. Smith and T.A. Wyatt, Development of draft rules for aerodynamic stability, in: Bridge aerodynamics (Institute Civil Engineering, Thomas Telford Ltd., London, 1981).

[3] A.W. Marris, A review on vortex streets, periodic wakes and induced vibration phenomena, J. Basic Eng. Trans. ASME (1964).

[4] E. Simiu and R.H. Scanlan, Wind effects on structures (Wiley, New York, 1978). [5] B.J. Vickery and R.I. Basu, Across-wind vibrations of structures of circular cross-section, J. Wind Eng.

lnd. Aerodyn. 12 (1983) 35 73. [-6] M.M. Zdravkovich, Scruton number, a proposal, J. Wind Eng. Ind. Aerodyn. 10 (1982) 263--265. [7] A. Larsen, A generalized model for assessment of vortex induced vibrations of flexible bridges, in:

Proc. 7th US Nat. Conf. on Wind engineering, 1993. [-8] D.E. Walshe, Some effects of turbulence on time-average wind-forces of sectional models of box girder

bridges (Bridge Aerodynamics, Institute Civil Engineering, Thomas Telford Ltd., London, 1981). [9] A.H. Nayfeh, Perturbation methods, Pure and applied mathematics series (Wiley, New York, 1973).

[10] A.F. Daly, Evaluation of methods of predicting the across-wind response of chimneys, CICIND report, Vol. 2, No. 1 (1986}.