advances in aeroelastic analyses of suspension and cable-stayed bridges

Journal of Wind Engineeringand Industrial Aerodynamics 74—76 (1998) 73—90

Advances in aeroelastic analysesof suspension and cable-stayed bridges

Allan Larsen*COWI Consulting Engineers and Planners AS, Parallelvej 15, DK-2800 Lyngby, Denmark

Abstract

Cross-section shape is an important parameter for the wind response and aeroelastic stabilityof long span suspension and cable-stayed bridges. Numerical simulation methods have nowbeen developed to a stage where assessment of the effect of practical cross-section shapes onbridge response is possible. The present paper reviews selected numerical simulations carriedout for a long-span suspension bridge using finite difference and discrete vortex methods.Comparison of simulations to existing wind tunnel data is discussed. Further, the paperaddresses the aerodynamics and structural response of four generic cross-section shapesdeveloped from the well-known plate girder section of the first Tacoma Narrows Bridge. Finallya case study involving the wind response of a 400 m main span cable-stayed bridge isdiscussed. ( 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Numerical methods; Bridge aerodynamics; Buffeting; Vortex shedding excitation;Aeroelastic instability

1. Introduction

The engineering discipline of bridge-aerodynamics was born of the spray risingfrom the fall of the first Tacoma Narrows Bridge into the Puget Sound in 1940. In hismonumental investigation of the bridge collapse, Farquharson [1] covered a widerange of technical aspects ranging from experimental techniques over aerodynamicsand structural dynamics to guidelines for bridge design. Probably, the singlemostimportant finding of the Tacoma Narrows investigation was that vortex-sheddingexcitation and flutter instability of a complete suspension bridge could be accuratelyrepresented by a spring-supported section model of the deck structure. This important

*E-mail: [email protected].

0167-6105/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved.PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 0 0 7 - 5

result was later corroborated by wind tunnel investigations conducted in the UnitedKingdom in connection with design of a suspension bridge for crossing of the riverSevern, Frazer and Scruton [2]. Further versatility was added to section model testingby Davenport [3], who advanced a method for assessment of buffeting response ofcomplete bridge structures to turbulent wind based on aerodynamic section data.Needless to say, section model testing offers substantial savings relative to testing offull aeroelastic bridge section models.

Structural analyses of bridges have moved from laboratory testing of physicalmodels to computer-based finite element modelling. This development has allowedthe designer to experiment with different structural systems and configurationswithout resorting to expensive and time-consuming physical testing. Aerodynamicanalysis of bridges has not seen a similar development due to the complexity of thefluid dynamic phenomena involved, hence most aerodynamic analyses of bridgestructures are still restricted by aerodynamic data obtained from wind tunnel testing.Numerical fluid dynamic models and computer capacity have developed over the pastdecade to a stage where the bridge designer may start to exploit these new techniquesin actual design work in much the same way as physical section model tests. Inparticular, numerical simulations appear well suited for design studies of the effect ofcross-section shape on bridge response to wind loading, thus presenting an efficienttool for weeding out inefficient cross-sections before embarking on confirmatory windtunnel testing.

The present paper will highlight some recent comparisons between wind tunnelsection model results and numerical simulations. The main body of the paper will bedevoted to a design study of the effect of cross-section shape on bridge response takingthe first Tacoma Narrows Bridge as an example. Finally the paper will outlinea numerical design study carried out for determination of the most favourablecross-section shape for a 400 m main span cable-stayed bridge.

2. Model tests and numerical simulations

The wind design of the East Bridge was carried out in the time span 1989—1992 andwas based on extensive wind tunnel section model testing. Since that time numericalmethods have developed to the extent that two-dimensional aerodynamic section datamay be calculated with acceptable accuracy for design studies. In order to illustrate thispoint, measured and calculated wind load coefficients for the girder cross-sections of theEast Bridge suspended spans and approach spans, shown in Fig. 1, will be compared.

2.1. East Bridge suspended spans

Steady-state wind loads for the cross-section of the East Bridge suspendedspans have been reported by two different workers using different numerical simu-lation techniques. Kuroda [5] applied a gird-based finite-difference method(FDM) using the pseudo-compressibility technique for solution of the incompressibletwo-dimensional Navier—Stokes equations at Reynolds number Re"3]105 (based

74 A. Larsen/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 73–90

Fig. 1. Girder cross-sections of the East Bridge suspended spans and approach spans.

Table 1Comparison of steady-state wind load coefficients obtained from numericalsimulations and wind tunnel testing of a 1 : 80 section model. East BridgeSuspended spans

Method CD0

(-) CL0

(-) CM0

(-) dCL/da (-/rad) dC

M/da (-/rad)

FDM [5] 0.071 !0.100 0.025 6.48 1.15DVM [6] 0.061 0.000 0.027 4.13 1.15Experiment 0.081 0.067 0.028 4.37 1.17

on cross-section width B). Walther [6] applied the grid-free discrete vortex method(DVM) for solution of the two-dimensional vorticity equation representing the flowaround the cross-section at Re"105. The wind loads reported are made non-dimensional through division with the dynamic head 1

2oº2 and section width B:

CD"

D12oº2B

, CL"

¸

12oº2B

, CM"

M12oº2B2

. (1)

Table 1 compares simulated wind load coefficients CD0

, CL0

, CM0

at zero angle ofattack and lift and moment slopes dC

L/da, dC

M/da to experimental values obtained

from wind tunnel testing of a 1 : 80 section model as reported by Larsen [4].Satisfactory agreement between simulations and experiment is demonstrated for

most of the coefficients with the exception of dCL/da obtained from the FDM

simulations. This coefficient is 48% in excess of the experimental data. In comparingC

D0values it shall be remembered that the physical section model was equipped with

light tubular railings and crash barriers, whereas the numerical geometry models onlyreproduced the gross trapezoidal cross-section shape. Simple calculations allowingeach of the railing components to be exposed to the free stream wind speed yielda drag contribution of *C

D0"0.023 which brings simulations and experiment in

better agreement.

2.2. East Bridge approach spans

Similar to the cross-section of the suspended spans the approach spans (Fig. 2,right) have been subject to two-dimensional numerical flow simulations. Selvam

A. Larsen/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 73–90 75

Table 2Comparison of drag coefficient and Strouhal num-ber obtained from numerical simulations and windtunnel testing of a 1 : 80 scale section model. EastBridge Approach spans

Method CD0

(-) St (-)

FDM — LES [7] 0.187 0.166—0.199DVM [8] 0.179 0.167Experiment 0.190 0.170

et al. [7] applied a finite-difference method including a large eddy simulation (LES)model for prediction of the important drag loading and Strouhal number andvortex-shedding frequency. Larsen and Walther [8] report similar results obtainedby means of the discrete vortex method. Table 2 offers a comparison betweennumerical simulations and experimental results of drag coefficient and Strouhalnumber:

CD"

D

12oº2B

, St"fH

º

, (2)

where H is cross-wind section depth (H"7.0 m) and f is vortex sheddingfrequency.

As in the case of the suspended span cross-section the numerical simulations are infair agreement with experimental data. Again railings and crash barriers were notincluded in the numerical models, hence slightly lower C

D0values are to be expected

when comparing with the experiment.Further comparisons between experiment and numerical simulations are presented

by Larsen and Walther [8] for the H-shaped cross-section of the first TacomaNarrows Bridge and for a twin-box cross-section developed for a fixed link across theStraits of Gibraltar. These results are equally promising indicating that numericalsimulations of flow around bridge girders are worth while in bridge design and retrofitstudies.

3. Models for bridge response to wind

Bridge response to wind is mainly governed by the aerodynamic properties of thegirder cross-sections, structural parameters such as mass, mass moment of inertia,eigenfrequencies and damping and for buffeting response the turbulence properties ofthe wind field. The appendix offers a brief run down of mathematical models whichmay be used for a first-order assessment of the three most important types ofwind-induced response: (1) along-wind buffeting response (drag direction), (2) vertical


vortex-shedding excitation and (3) critical wind speed for onset of flutter. Besidesbeing of practical use, the models illustrate that bridge buffeting, vortex-sheddingexcitation and flutter stability may be calculated, once drag coefficient C

D0, root mean

square lift coefficient CRMSL

, Strouhal number St and aerodynamic derivatives H*1..4

,A*

1..4are available for a given bridge girder cross-section. A more complete descrip-

tion of the use of aerodynamic cross-section data in bridge response analyses is offeredby Scanlan [12]. The following section will focus on the above-mentioned aerody-namic properties and simulation by the discrete vortex method.

4. Discrete vortex method for 2D bridge deck cross-sections

A distinct feature of flow about bluff bodies, stationary or in time-dependentmotion, is the shedding of vorticity in the wake which balances the change of fluidmomentum along the body surface. The vorticity shed at an instant in time isconvected downstream by the mean wind speed but continues to affect the aerody-namic loads on the body. A mathematical model for the flow around bluff bodies wasdeveloped within the framework of the discrete vortex method and programmed forcomputer by Walther [6]. The resulting numerical code DVMFLOW establishesa “grid-free” time-marching simulation of the vorticity equation well suited forsimulation of 2D bluff body flows. An outline of the mathematical model and thesimulated flow about a flat plate is presented by Walther and Larsen [9]. The input toDVMFLOW simulations is a boundary panel model of the cross-section contour. Theoutput of DVMFLOW simulations is time progressions of surface pressures andsection loads (drag, lift and moment). In addition, maps of the flow field (vectorplots), vortex positions and streamlines at prescribed time steps are available.Steady-state wind load coefficients and Strouhal number are obtained from timeaverages and frequency analysis of simulated loads on stationary panel models.Aerodynamic derivatives are obtained from post-processing of simulated timeseries of forced harmonic motion as detailed by Larsen [10] in a numerical investiga-tion of five generic bridge deck cross-sections tested in a wind tunnel by Scanlan andTomko [11].

5. Five Bridge deck cross-sections — an example

The lively wind response (the galloping) and the final collapse of the first TacomaNarrows Bridge, Fig. 2, was established to be due to the aerodynamically unfavorablecross-section shape and lightness of the bridge structure.

Although a number of investigations has pointed out that H-shaped cross-sectionssimilar to the first Tacoma Narrows are undesirable from an aerodynamic point ofview they remain attractive from an economic point of view as well as due to ease offabrication. The present example will thus consider the aerodynamic effect of foursimple modifications of the parent H-shaped cross-section.


Fig. 2. First Tacoma Narrows Bridge. Girder deck cross-section, elevation and structural data applicableto the first asymmetric mode of vibration [1].

Fig. 3. Cross-section shapes considered in the present study.

5.1. Cross-sections investigated

The five cross-sections investigated are shown in Fig. 3.The parent cross-section denoted H is a slightly simplified version of the first

Tacoma Narrows deck omitting cross-girders and curbs but reproducing the longitu-dinal edge girders and floor slab. Section C (channel type) is obtained from theH section simply by adding a top plate. Section R (rectangular type) is obtained byadding a bottom-plate to the C section. The CE section (channel/edge) is obtained


from the C section by adding triangular edge-fairings to the C section. Finally theB section (box type) is obtained by adding the triangular edge-fairings to the R sectionor closing the bottom of the CE section. All dimensions in Fig. 3 are referred to thewidth B of the parent H section. The circumference of each cross-section wassubdivided into a total of 300 surface vortex panels. The discretisation allowed flow atReynolds number Re"105 (Re"ºB/l) to be simulated.

5.2. Simulation of flow about stationary sections

Drag coefficient CD0

, root mean square lift coefficient CRMSL

and Strouhal number Stare obtained from simulations of the flow about the five cross-sections fixed in space.An angle of attack of 0° of the wind flow was assumed (angle between flow directionand section chord). Each simulation was run for 30 non-dimensional time units¹"tº/B where t is the time, º is the flow speed and B is the chord length.A non-dimensional time increment *¹"0.025 was adopted throughout the simula-tions. At each time step the cross-section surface pressure distribution was computedfrom the local flux of surface velocity. The section surface pressures were finallyintegrated along the contour to form time traces of the section drag D and lift ¸ forces.Lastly, the computed aerodynamic forces were expressed in non-dimensional formfollowing Eq. (1). Fig. 4 shows an example of the simulated time traces of C

D0and

CL

obtained for cross-section H.The C

Dtrace displays initial very high values associated with the instantaneous

start up of the flow simulation. After an exponential decay the CD

trace settles arounda mean value C

D0"0.28 after approximately 5 non-dimensional time units.

CD0

"0.28 is in satisfactory agreement with CD0

"0.29!0.30 reported by Far-quharson [1]. The C

Ltrace develops very distinct oscillations with period ¹+1.7

associated with formation of vortex roll up in the wake — the well-known von Karmanvortex street, Fig. 5. The wake pattern obtained from simulation of the flow

Fig. 4. Simulated time traces of drag coefficient CD

and lift coefficient CL

for cross-section H.


Fig. 5. Formation of von Karman vortex street in the wake of cross-section H.

Fig. 6. Formation of von Karman vortex street in the wake of cross-section B.

Fig. 7. Simulated time traces of drag coefficient CD

and lift coefficient CL

for cross-section B.

around cross-section B is shown in Fig. 6, whereas simulated time traces of CD

andC

Lare shown in Fig. 7. A summary of aerodynamic data for all sections is given in

Table 1.From Table 3 it is noted that the closed-box section B displays better aerodynamic

performance than the remaining cross-sections, i.e. lower CD

and CRMSL

. The parentcross-section H appears to yield the worst aerodynamic performance.


Table 3Flow fields in the vicinity of the cross sections and C

D0, CRMS

Land St values extracted

Cross section geometry and flow patterns CD0

CRMSL

St

0.28 0.37 0.11

0.23 0.33 0.11

0.23 0.24 0.09

0.16 0.34 0.09

0.11 0.17 0.13


By comparing Figs. 4 and 7 it is noted that the drag loading and the oscillating lifthave decreased considerably from section H to section B due to the geometricmodifications introduced.

5.3. Motion-dependent aerodynamic forces — aerodynamic derivatives

Simulations of forced harmonic bending and twisting section motions are usedfor determination of motion-induced aerodynamic forces. Lift and moment timetraces obtained from numerical simulations are processed to yield aerodynamicderivatives following the procedure presented by Larsen and Walther [10]. The G5and G2 cross-sections investigated in Ref. [10] are almost identical to the Hand B sections considered above, hence the simulated G5, G2 aerodynamic deriva-tives will be considered representative of the present H and B cross-sections andthus reviewed here. Fig. 8 superimposes the aerodynamic derivatives obtained fromsimulations on the wind tunnel data presented by Scanlan and Tomko [11]. Onlythe six major derivatives are reported in line with Scanlan and Tomko’s work.The remaining H*

4and A*

4derivatives are of little significance for practical flutter

predictions.A few comments are appropriate at this point. The simulated H*

1!H*

3and the

A*1

derivatives representative of the B section compare very well to the airfoil data (A).The A*

2and A*

3display less correlation with the airfoil data, possibly due to leading

edge separation caused by the sharp-edged corners of the bridge sections. Whencomparing the simulations to the experimental bridge section data it is noticed thatH*

2obtained from simulations is all together different. The A*

2and A*

3derivatives are,

however, in very good agreement with the derivatives measured for the bridge deckmodel. In case of the H section the A*

2derivative is the most important coefficient as

its change of sign (from negative at low reduced wind speeds to positive at high windspeeds) signifies one-degree-of-freedom torsional flutter. The simulations which arerun at a forced twisting amplitude of 3° indicate a cross-over point for A*

2at about

twice the wind speed as compared to the wind tunnel data. The remaining aero-dynamic derivatives for the H cross-section are in reasonable agreement with theexperiments.

6. Influence of cross-section shape on wind response and stability

The role of the section shape-dependent aerodynamic parameters on horizontalalong-wind buffeting response, vertical vortex-induced response and critical windspeed for onset of flutter is illustrated through the set of expressions given in theappendix. Assuming similar structural properties and wind conditions for bridgesinvolving the five cross-sections investigated allows assessment of their relativeresponse to wind. Table 2 presents such an evaluation based on the C

D0, CRMS

Land St

coefficients summarised in Table 3 and the simulated aerodynamic derivatives givenin Fig. 8. Cross-section H serves as reference and is assigned a unit response (1.0),


Fig. 8. Comparison of simulated aerodynamic derivatives for the H and B cross-sections aerodynamicderivatives obtained from wind tunnel section model tests.

whereas the remaining section responses are evaluated relative to this. Critical windspeeds º

#for onset of flutter are calculated specifically in m/s using the structural data

of Fig. 2.It is noted from Table 4 that wind-induced response due to along-wind buffeting

and vortex shedding is sensitive to the cross-section shape. The trapezoidal boxsection appears to be significantly less susceptible to wind response than the remaining


Table 4Relative wind-induced response of bridges due to different cross-sectionshapes

Cross-sectiondesignation

Horizontalbuffeting response

Vertical vortexresponse

Critical wind speedº

#(m/s)

H 1.0 1.0 11.5C 0.82 0.89 —R 0.82 0.97 —CE 0.76 1.42 —B 0.53 0.34 20.5

cross-sections. Also the flutter performance of trapezoidal box cross-section appearsto be superior at least to the notoriously unstable H-shaped deck cross-section.

7. Case study — wind response of a 400 m main span cable-stayed bridge

Numerical simulations using the discrete vortex method are run on a regular basisby the author’s company for assessment of the aerodynamic performance of newbridge projects or retrofits. The present case study considers a 400 m main spancable-stayed bridge, Fig. 9.

The bridge was tendered with two alternative cross-sections: (A1) A compositecross-section composed of a concrete deck slab carried by rectangular box edgebeams, plate cross-girders and longitudinals all in steel. (A2) A composite cross-section composed of a concrete deck slab supported by a closed steel box struc-ture. Both alternatives were equipped with solid New Jersey type crash barrierswhich caused some concern with respect to the aerodynamic performance of thebridge. DVMFLOW simulations of steady-state wind load coefficients at !3°, 0°and 3° angle of attack were carried out for application to buffeting calculations andfor identification of the lock-in wind speed for vortex-shedding excitation. Simula-tions of motion-induced aerodynamic loads were carried out to obtain aerodynamicderivatives for input to flutter routines. Vertical vortex-induced responses weresimulated directly in DVMFLOW by allowing the cross-sections to be supported byvertical spring elements tuned to the lock-in frequency. Simulated flow fields aboutthe alternative plate girder and box girder cross-sections are shown in Fig. 10.

The flow about the plate girder cross-section forms large recirculating vorticalstructures below the deck in the compartments between the edge girders and thelongitudinals. In contrast, the flow about the box girder cross-section is smooth alongthe slightly curved bottom plate. The differences in the respective flow fields carry overin the predicted aerodynamic properties and the bridge response as summarised inTable 5.

For the present practical example it is noted that the box section A2 is aero-dynamically superior to the plate section A1. The drag coefficient and the vertical


Fig. 9. 400 m main span cable-stayed bridge studied by discrete vortex simulations.

Fig. 10. Simulated flow about alternative girder cross-sections for a 400 m cable-stayed bridge.

Table 5Predicted aerodynamic drag C

D0, Strouhal number St, vertical vortex-induced response h (in first vertical

bending mode), flutter mode and critical wind speed for onset of flutter º#

Cross-sectiondesignation

Drag coefficientC

D0

Strouhal no.St

Vertical vortexresponse h (m)

Flutter mode Flutter wind speedº

#(m/s)

Plate section, A1 0.12 0.13 0.055 1DOF 130Box section, A2 0.07 0.16 0.034 2DOF 210

vortex-induced response of the A2 cross-section is reduced by approximately 40%relative to the A1 cross-section, whereas the critical wind speed for onset of flutter isincreased by 60%.


8. Conclusion

The present paper has discussed the use of two-dimensional numerical flow simula-tions in bridge aerodynamics. The paper has focused on the expected accuracy,correlation with wind tunnel test results and the application in bridge design studiesfor determination of cross-section shape on bridge response to wind. It is concludedthat numerical simulations present a worthwhile alternative to section model testingin cases where the deck cross-section geometry investigated allow meaningful dis-cretisation in two dimensions. From a bridge designers point of view numericalsimulations appear as an efficient tool for weeding out inefficient cross-sectionalternatives before embarking on confirmatory wind tunnel testing.

Appendix A. Mathematical models of bridge response to wind

A.1. Horizontal buffeting response to turbulent winds

The buffeting theory developed by Davenport [3] considers each vibration modereceiving excitation by atmospheric turbulence as a one-degree-of-freedom (1DOF)oscillator. In this format root mean square bridge response p

xat the eigenfrequency

f of each individual horizontal mode of oscillation may be obtained as

px"C

D0 AoB3

M*B Aº

fBB2 1

8S1

p3(f4#f

!)IuS f S

u( f )

p2u

L

P0

e~C@s~s{@f@Uu(s)

L

P0

u(s@) ds ds@,

(A.1)

where f4and f

!are the structural and aerodynamic damping levels relative to critical,

M*"m:L0u2(s) ds is the modal mass in the mode of motion considered, m is the

mass/unit length of structure, Iuis the (along-wind) turbulence intensity, fS

u( f )/p2

uis

the normalised power spectrum of turbulence, J:L0e~C@s~{@f@Uu(s):L

0u(s@) ds ds@ is the

spanwise joint acceptance function, C+4—8, and u(s), s and ¸ are mode shapespanwise coordinate and span length.

The important thing to notice at this point is that the horizontal buffeting responseis directly proportional to the drag coefficient C

D0which again is a function of the

cross-section shape. The remaining parameters relate either to the structural proper-ties of the bridge (modal mass and mode shape) or to the wind climatic conditionsprevailing at the bridge site.

A.2. Vertical vortex-induced response

Vertical vortex-induced response of bridge structures may be treated in a modalformat much the same way as the resonant horizontal buffeting response given above.Wyatt and Scruton [13] have proposed a 1DOF oscillator model in which the vertical


periodic load due to vortex shedding excitation is represented by a root mean squarelift coefficient CRMS

Lto be obtained from wind tunnel section model tests. A slightly

modified version of the Wyatt/Scruton model yields the following root mean squarevertical vortex shedding response p

h:

ph"A

CRMSL

St2 B A1

16p2(f4#f

!)B A

oBH2

m B:L0Du(s)D ds

:L0u2(s) ds

, (A.2)

where m is the cross-section mass/unit length and :L0Du(s)D ds/:L

0u2(s) ds is a factor '1

accounting for the effect of mode shape.It is noticed that the bridge response, at least to a first approximation, is propor-

tional to the ratio of the root means square lift coefficient to the Strouhal numbersquared, items which again are functions of the cross-section shape. The remainingparameters relate to the structural properties of the bridge.

A.3. Aerodynamic damping

A measure for the aerodynamic damping f!

(relative to critical) is needed forcarrying out response calculations for along-wind buffeting and cross-wind vortex-shedding excitation. In the case of along-wind buffeting, the aerodynamic dampingarises from a force opposing the motion in the direction of the mean wind. In this case,f!is expressed in terms of the cross section drag coefficient:

f!"

oºBCD0

4pm. (A.3)

In the case of vertical vortex-shedding excitation, the aerodynamic damping arisingfrom a cross-wind force opposing the vertical motion is negative. In applying theaerodynamic derivative formulation of motion-induced aerodynamic forces, Scanlan[12] the cross-wind aerodynamic damping is obtained as

f!"!

oB2H*1

2m. (A.4)

A.4. Aeroelastic instability – flutter

Two types of flutter instabilities are commonly encountered in bridge engineering:(1) 1DOF torsional flutter by which the girder responds to motion induced aerody-namic forces in a pure torsional mode. (2) 2DOF flutter by which the bridge girderresponds in a combined bending and torsional mode due to cross-coupled motion-induced aerodynamic forces. Mathematical models for onset of one or two-degree-of-freedom flutter instability are developed from similar modal concepts as the modelsfor buffeting and vortex shedding response. The representation of the motion inducedaerodynamic forces acting on a cross section is however slightly more complicated.A convenient framework for distinction of flutter type (one or two-degree-of-freedom)


and prediction the critical wind speed for onset of flutter is given by Scanlan [12], whointroduced a set of aerodynamic coefficients, the so-called aerodynamic derivatives,representing the motion induced aerodynamics of a given cross section. The aerody-namic derivatives which are found to be functions of reduced wind speed º/fB andcross section geometry may be related to the well known lift and moment coefficientsin cases where the cross-section is undergoing forced harmonic bending or twistingmotion [11].

A.5. One-degree-of-freedom torsional flutter

A mathematical model for pure torsional flutter is presented by the following1DOF oscillator assuming time complex harmonic twisting motion of the crosssection (i is the imaginary unit):

I[(u2a!u2)#i2f4uau]a"oº2B2 A

uB

º B2

[iA*2#A*

3]a, (A.5)

where I is the cross-section mass moment of inertia/unit length, ua and u are circulareigenfrequency and circular frequency of motion and A*

2and A*

3are aerodynamic

derivatives representing aerodynamic damping and stiffness.The critical wind speed for onset of 1DOF torsional flutter is identified as the wind

speed where the structural damping balances “negative” aerodynamic damping. Fromthe equation of motion this condition is fulfilled for the following critical value of(A*

2)#:

(A*2)#"

2If4

oB4(A.6)

taking ua+u.If A*

2is plotted in a diagram as function of º/fB as is common practice, the critical

wind speed for onset of flutter is obtained as the abscissa (º/fB)#to (A*

2)#. Aerodynam-

ically speaking, 1DOF torsional flutter is distinguishable from 2DOF flutter by thefact that the A*

2aerodynamic derivative (which is proportional to the aerodynamic

damping in torsion) changes sign from negative at low º/fB to positive at some highervalue of º/fB.

A.6. Two-degree-of-freedom coupled flutter

Cross-sections for which the A*2

aerodynamic derivative remains negative for allreduced wind speeds º/fB (A*

2negative"positive aerodynamic damping in torsion)

are likely to display 2DOF coupled vertical/torsional flutter behaviour. This occurs atthe wind speed where the motion-induced aerodynamic loads cause vertical andtorsional frequencies of motion to collapse into one common frequency. A mathemat-ical model for coupled vertical/torsional flutter is presented by the following set ofone-degree-of-freedom oscillators assuming time complex harmonic vertical (h) and


twisting (a) motion of the cross-section:

m[(u2h!u2)#i2u

huf]h"oº2BA

uB

º B2

C(iH*1#H*

4)h

B#(iH*

2#H*

3)aD,

(A.7)

I[(u2a!u2)#i2uauf]a"oº2B2AuB

º B2

C(iA*1#A*

4)h

B#(iA*

2#A*

3)aD,

(A.8)

where uh

and ua are the respective circular eigenfrequencies, u a common flutterfrequency and H*

124, A*

124are the self-induced section loads — the aerodynamic

derivatives.It is noted that the aerodynamic loads introduces coupling between the equations

for the vertical (h) and twisting (a) motion. Introducing the frequency ratio c"ua/uhand the frequency ratio X"u/u

hand rearranging the equations of motion yields the

flutter determinant which, when set equal to zero defines the flutter point:

KK1!X2!

oB2

mH*

4X2#iA2fX!

oB2

mH*

1X2B !

oB2

mH*

3X2!

oB2

mH*

2X2

!

oB4

IA*

4X2!i

oB4

IA*

1X2 c2!X2!

oB4

IA*

3X2#iA2cXf!

oB4

IA*

2X2B KK"0.

The flutter determinant defines a fourth-order real and a third-order imaginaryalgebraic equation to be solved for X introducing the H*

124, A*

124coefficients

obtained at successive values of the reduced wind speed º/fB. Onset of 2DOF flutterwill occur at the particular reduced wind speed (º/fB)

#where the roots of the real and

imaginary equations X3%!-, X*.!' are identical"X#. Finally, the critical wind speed

º#

for onset of 2DOF coupled flutter is obtained as

º#"A

º

fBB#

f)BX

#. (A.9)

For a more detailed and complete description of the use of aerodynamic cross-section data in bridge response analyses the reader is referred to the state-of-the-art-review by Scanlan [12].

References

[1] F.B. Farquharson, Aerodynamic stability of suspension bridges, University of Washington Experi-mental Station, Bull. 116, Part I—V, 1949—54.

[2] R.A. Frazer, C. Scruton, A summarised account of the severn bridge aerodynamic investigation, NPLAero Report, 222, London, HMSO, 1952.

[3] A.G. Davenport, Buffeting of a suspension bridge by storm winds, J. Struct. Div. ASCE (1962)233—264.

[4] A. Larsen, Aerodynamic aspects of the final design of the 1624 m suspension bridge across the greatbelt, J. Wind Eng. Ind. Aerodyn. 48 (1993) 261—285.


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advances in aeroelastic analyses of suspension and cable-stayed bridges

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