aero last i c analysis

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Journal of Wind Engineering

and Industrial Aerodynamics 7476 (1998) 7390

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 stability

of long span suspension and cable-stayed bridges. Numerical simulation methods have now

been developed to a stage where assessment of the effect of practical cross-section shapes on

bridge response is possible. The present paper reviews selected numerical simulations carried

out 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 paper

addresses the aerodynamics and structural response of four generic cross-section shapes

developed from the well-known plate girder section of the first Tacoma Narrows Bridge. Finally

a case study involving the wind response of a 400 m main span cable-stayed bridge is

discussed. 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 rising

from the fall of the first Tacoma Narrows Bridge into the Puget Sound in 1940. In his

monumental investigation of the bridge collapse, Farquharson [1] covered a wide

range of technical aspects ranging from experimental techniques over aerodynamics

and structural dynamics to guidelines for bridge design. Probably, the singlemost

important finding of the Tacoma Narrows investigation was that vortex-sheddingexcitation and flutter instability of a complete suspension bridge could be accurately

represented 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


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result was later corroborated by wind tunnel investigations conducted in the United

Kingdom in connection with design of a suspension bridge for crossing of the river

Severn, Frazer and Scruton [2]. Further versatility was added to section model testing

by Davenport [3], who advanced a method for assessment of buffeting response of

complete bridge structures to turbulent wind based on aerodynamic section data.Needless to say, section model testing offers substantial savings relative to testing of

full aeroelastic bridge section models.

Structural analyses of bridges have moved from laboratory testing of physical

models to computer-based finite element modelling. This development has allowed

the designer to experiment with different structural systems and configurations

without resorting to expensive and time-consuming physical testing. Aerodynamic

analysis of bridges has not seen a similar development due to the complexity of the

fluid dynamic phenomena involved, hence most aerodynamic analyses of bridge

structures are still restricted by aerodynamic data obtained from wind tunnel testing.

Numerical fluid dynamic models and computer capacity have developed over the past

decade to a stage where the bridge designer may start to exploit these new techniques

in actual design work in much the same way as physical section model tests. In

particular, numerical simulations appear well suited for design studies of the effect of

cross-section shape on bridge response to wind loading, thus presenting an efficient

tool for weeding out inefficient cross-sections before embarking on confirmatory wind

tunnel testing.

The present paper will highlight some recent comparisons between wind tunnelsection model results and numerical simulations. The main body of the paper will be

devoted to a design study of the effect of cross-section shape on bridge response taking

the first Tacoma Narrows Bridge as an example. Finally the paper will outline

a numerical design study carried out for determination of the most favourable

cross-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 19891992 and

was based on extensive wind tunnel section model testing. Since that time numerical

methods have developed to the extent that two-dimensional aerodynamic section data

may be calculated with acceptable accuracy for design studies. In order to illustrate this

point, measured and calculated wind load coefficients for the girder cross-sections of the

East 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 suspended

spans 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 incompressible

two-dimensional NavierStokes equations at Reynolds number Re"310 (based

74 A. Larsen/J. Wind Eng. Ind. Aerodyn. 7476 (1998) 7390


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

Table 1

Comparison of steady-state wind load coefficients obtained from numerical

simulations and wind tunnel testing of a 1 : 80 section model. East Bridge

Suspended spans

Method C

(-) C

(-) C

(-) dC

/d (-/rad) dC

/d (-/rad)

FDM [5] 0.071 !0.100 0.025 6.48 1.15

DVM [6] 0.061 0.000 0.027 4.13 1.15

Experiment 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"10. The wind loads reported are made non-

dimensional through division with the dynamic head and section width B:

C"

D

B

, C"

B

, C"

M

B

. (1)

Table 1 compares simulated wind load coefficients C

, C

, C

at zero angle of

attack and lift and moment slopes dC

/d, dC

/d to experimental values obtainedfrom 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 dC

/d obtained from the FDMsimulations. This coefficient is 48% in excess of the experimental data. In comparingC

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

light tubular railings and crash barriers, whereas the numerical geometry models only

reproduced the gross trapezoidal cross-section shape. Simple calculations allowing

each of the railing components to be exposed to the free stream wind speed yield

a drag contribution ofC"0.023 which brings simulations and experiment inbetter 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. 7476 (1998) 7390 75


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Table 2

Comparison of drag coefficient and Strouhal num-

ber obtained from numerical simulations and wind

tunnel testing of a 1 : 80 scale section model. East

Bridge Approach spans

Method C

(-) St (-)

FDMLES [7] 0.187 0.1660.199

DVM [8] 0.179 0.167

Experiment 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 and

vortex-shedding frequency. Larsen and Walther [8] report similar results obtained

by means of the discrete vortex method. Table 2 offers a comparison between

numerical simulations and experimental results of drag coefficient and Strouhal

number:

C"

D

B

, St"fH

, (2)

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

frequency.

As in the case of the suspended span cross-section the numerical simulations are in

fair agreement with experimental data. Again railings and crash barriers were not

included in the numerical models, hence slightly lowerC

values 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 the

Straits of Gibraltar. These results are equally promising indicating that numerical

simulations of flow around bridge girders are worth while in bridge design and retrofit

studies.

3. Models for bridge response to wind

Bridge response to wind is mainly governed by the aerodynamic properties of the

girder cross-sections, structural parameters such as mass, mass moment of inertia,

eigenfrequencies and damping and for buffeting response the turbulence properties of

the wind field. The appendix offers a brief run down of mathematical models which

may be used for a first-order assessment of the three most important types of

wind-induced response: (1) along-wind buffeting response (drag direction), (2) vertical



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vortex-shedding excitation and (3) critical wind speed for onset of flutter. Besides

being of practical use, the models illustrate that bridge buffeting, vortex-shedding

excitation and flutter stability may be calculated, once drag coefficientC

, root mean

square lift coefficient C

, Strouhal number St and aerodynamic derivativesH*

,

A* are 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 offered

by 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 fluid

momentum along the body surface. The vorticity shed at an instant in time is

convected 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 was

developed within the framework of the discrete vortex method and programmed for

computer by Walther [6]. The resulting numerical code DVMFLOW establishes

a grid-free time-marching simulation of the vorticity equation well suited for

simulation of 2D bluff body flows. An outline of the mathematical model and the

simulated flow about a flat plate is presented by Walther and Larsen [9]. The input to

DVMFLOW simulations is a boundary panel model of the cross-section contour. The

output of DVMFLOW simulations is time progressions of surface pressures and

section loads (drag, lift and moment). In addition, maps of the flow field (vector

plots), vortex positions and streamlines at prescribed time steps are available.

Steady-state wind load coefficients and Strouhal number are obtained from time

averages and frequency analysis of simulated loads on stationary panel models.

Aerodynamic derivatives are obtained from post-processing of simulated time

series 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 and

Tomko [11].

5. Five Bridge deck cross-sectionsan example

The lively wind response (the galloping) and the final collapse of the first Tacoma

Narrows 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-sections

similar to the first Tacoma Narrows are undesirable from an aerodynamic point of

view they remain attractive from an economic point of view as well as due to ease of

fabrication. The present example will thus consider the aerodynamic effect of four

simple modifications of the parent H-shaped cross-section.



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Fig. 2. First Tacoma Narrows Bridge. Girder deck cross-section, elevation and structural data applicable

to 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 the

H section simply by adding a top plate. Section R (rectangular type) is obtained by

adding a bottom-plate to the C section. The CE section (channel/edge) is obtained



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from the C section by adding triangular edge-fairings to the C section. Finally the

B section (box type) is obtained by adding the triangular edge-fairings to the R section

or closing the bottom of the CE section. All dimensions in Fig. 3 are referred to the

width B of the parent H section. The circumference of each cross-section was

subdivided into a total of 300 surface vortex panels. The discretisation allowed flow atReynolds number Re"10 (Re"B/) to be simulated.

5.2. Simulation of flow about stationary sections

Drag coefficientC

, root mean square lift coefficient C

and Strouhal number St

are obtained from simulations of the flow about the five cross-sections fixed in space.

An angle of attack of 0of the wind flow was assumed (angle between flow direction

and section chord). Each simulation was run for 30 non-dimensional time units"t/B where t is the time, is the flow speed and B is the chord length.

A non-dimensional time increment"0.025 was adopted throughout the simula-tions. At each time step the cross-section surface pressure distribution was computed

from the local flux of surface velocity. The section surface pressures were finally

integrated along the contour to form time traces of the section dragDand lift forces.

Lastly, the computed aerodynamic forces were expressed in non-dimensional form

following Eq. (1). Fig. 4 shows an example of the simulated time traces ofC

and

C

obtained for cross-sectionH.

The C trace displays initial very high values associated with the instantaneousstart up of the flow simulation. After an exponential decay the C

trace settles around

a mean value C"0.28 after approximately 5 non-dimensional time units.

C"0.28 is in satisfactory agreement with C

"0.29!0.30 reported by Far-

quharson [1]. The C

trace develops very distinct oscillations with period +1.7

associated with formation of vortex roll up in the wakethe well-known von Karman

vortex street, Fig. 5. The wake pattern obtained from simulation of the flow

Fig. 4. Simulated time traces of drag coefficientC

and lift coefficientC

for cross-section H.



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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 coefficientC

and lift coefficient C

for cross-section B.

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

and

C

are 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 C

and C

. The parent

cross-section H appears to yield the worst aerodynamic performance.



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Table 3

Flow fields in the vicinity of the cross sections and C

, C

and St values extracted

Cross section geometry and flow patterns C

C

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



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By comparing Figs. 4 and 7 it is noted that the drag loading and the oscillating lift

have decreased considerably from section H to section B due to the geometric

modifications introduced.

5.3. Motion-dependent aerodynamic forces aerodynamic derivatives

Simulations of forced harmonic bending and twisting section motions are used

for determination of motion-induced aerodynamic forces. Lift and moment time

traces obtained from numerical simulations are processed to yield aerodynamic

derivatives following the procedure presented by Larsen and Walther [10]. The G5

and G2 cross-sections investigated in Ref. [10] are almost identical to the H

and 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 from

simulations on the wind tunnel data presented by Scanlan and Tomko [11]. Only

the six major derivatives are reported in line with Scanlan and Tomkos work.

The remaining H*

and A*

derivatives are of little significance for practical flutter

predictions.

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

and the

A*

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

TheA*

andA*

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

edge separation caused by the sharp-edged corners of the bridge sections. When

comparing the simulations to the experimental bridge section data it is noticed thatH*

obtained from simulations is all together different. TheA*

andA*

derivatives are,

however, in very good agreement with the derivatives measured for the bridge deck

model. In case of the H section theA*

derivative is the most important coefficient as

its change of sign (from negative at low reduced wind speeds to positive at high wind

speeds) signifies one-degree-of-freedom torsional flutter. The simulations which are

run at a forced twisting amplitude of 3 indicate a cross-over point for A*

at 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 the

experiments.

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

The role of the section shape-dependent aerodynamic parameters on horizontal

along-wind buffeting response, vertical vortex-induced response and critical windspeed for onset of flutter is illustrated through the set of expressions given in the

appendix. Assuming similar structural properties and wind conditions for bridges

involving the five cross-sections investigated allows assessment of their relative

response to wind. Table 2 presents such an evaluation based on the C

,C

and St

coefficients summarised in Table 3 and the simulated aerodynamic derivatives given

in Fig. 8. Cross-section H serves as reference and is assigned a unit response (1.0),



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Fig. 8. Comparison of simulated aerodynamic derivatives for the H and B cross-sections aerodynamic

derivatives obtained from wind tunnel section model tests.

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

speedsfor 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 box

section appears to be significantly less susceptible to wind response than the remaining



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Table 4

Relative wind-induced response of bridges due to different cross-section

shapes

Cross-section

designation

Horizontal

buffeting response

Vertical vortex

response

Critical wind speed

(m/s)

H 1.0 1.0 11.5

C 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 studywind response of a 400 m main span cable-stayed bridge

Numerical simulations using the discrete vortex method are run on a regular basis

by the authors company for assessment of the aerodynamic performance of new

bridge 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 composite

cross-section composed of a concrete deck slab carried by rectangular box edge

beams, 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 barriers

which caused some concern with respect to the aerodynamic performance of the

bridge. DVMFLOW simulations of steady-state wind load coefficients at !3, 0

and 3 angle of attack were carried out for application to buffeting calculations and

for identification of the lock-in wind speed for vortex-shedding excitation. Simula-

tions of motion-induced aerodynamic loads were carried out to obtain aerodynamic

derivatives for input to flutter routines. Vertical vortex-induced responses were

simulated directly in DVMFLOW by allowing the cross-sections to be supported by

vertical spring elements tuned to the lock-in frequency. Simulated flow fields about

the alternative plate girder and box girder cross-sections are shown in Fig. 10.

The flow about the plate girder cross-section forms large recirculating vortical

structures below the deck in the compartments between the edge girders and thelongitudinals. In contrast, the flow about the box girder cross-section is smooth along

the slightly curved bottom plate. The differences in the respective flow fields carry over

in the predicted aerodynamic properties and the bridge response as summarised in

Table 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



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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 5

Predicted aerodynamic drag C

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

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

Cross-section

designation

Drag coefficient

C

Strouhal no.

St

Vertical vortex

responseh (m)

Flutter mode Flutter wind speed

(m/s)

Plate section, A1 0.12 0.13 0.055 1DOF 130

Box 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 is

increased by 60%.



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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 concluded

that numerical simulations present a worthwhile alternative to section model testing

in cases where the deck cross-section geometry investigated allow meaningful dis-

cretisation in two dimensions. From a bridge designers point of view numerical

simulations appear as an efficient tool for weeding out inefficient cross-section

alternatives 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 mode

receiving excitation by atmospheric turbulence as a one-degree-of-freedom (1DOF)

oscillator. In this format root mean square bridge response

at the eigenfrequencyfof each individual horizontal mode of oscillation may be obtained as

"C

BM*

fB 1

8 1

(#

)If S(f)

e(s)

(s) dsds,

(A.1)

whereand

are the structural and aerodynamic damping levels relative to critical,

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

mass/unit length of structure,Iis the (along-wind) turbulence intensity, fS(f)/ isthe normalised power spectrum of turbulence,

e(s)

(s) dsds is the

spanwise joint acceptance function, C+48, and (s), s and are mode shape

spanwise coordinate and span length.

The important thing to notice at this point is that the horizontal buffeting response

is directly proportional to the drag coefficient C

which 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 conditions

prevailing at the bridge site.

A.2. Verticalvortex-induced response

Vertical vortex-induced response of bridge structures may be treated in a modal

format 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



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periodic load due to vortex shedding excitation is represented by a root mean square

lift coefficient C

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

modified version of the Wyatt/Scruton model yields the following root mean square

vertical vortex shedding response:

"

C

St 1

16(#

)

BHm

(s)ds

(s) ds

, (A.2)

wheremis the cross-section mass/unit length and(s)ds/

(s) dsis 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 number

squared, items which again are functions of the cross-section shape. The remaining

parameters relate to the structural properties of the bridge.

A.3. Aerodynamic damping

A measure for the aerodynamic damping

(relative to critical) is needed for

carrying out response calculations for along-wind buffeting and cross-wind vortex-

shedding excitation. In the case of along-wind buffeting, the aerodynamic damping

arises from a force opposing the motion in the direction of the mean wind. In this case,

is expressed in terms of the cross section drag coefficient:

"

BC

4m . (A.3)

In the case of vertical vortex-shedding excitation, the aerodynamic damping arising

from a cross-wind force opposing the vertical motion is negative. In applying the

aerodynamic derivative formulation of motion-induced aerodynamic forces, Scanlan

[12] the cross-wind aerodynamic damping is obtained as

"!

BH*

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 models

for buffeting and vortex shedding response. The representation of the motion induced

aerodynamic 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)



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and prediction the critical wind speed for onset of flutter is given by Scanlan [12], who

introduced 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 /fB and

cross section geometry may be related to the well known lift and moment coefficientsin cases where the cross-section is undergoing forced harmonic bending or twisting

motion [11].

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

A mathematical model for pure torsional flutter is presented by the following

1DOF oscillator assuming time complex harmonic twisting motion of the cross

section (i is the imaginary unit):

I[(!)#i2]"B B

[i A*

#A*

], (A.5)

whereIis the cross-section mass moment of inertia/unit length, andare circulareigenfrequency and circular frequency of motion and A*

and A*

are 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 negativeaerodynamic damping. Fromthe equation of motion this condition is fulfilled for the following critical value of

(A*

):

(A*

)"

2I

B (A.6)

taking+.IfA*

is plotted in a diagram as function of/fBas is common practice, the critical

wind speed for onset of flutter is obtained as the abscissa (/fB)to (A*). Aerodynam-ically speaking, 1DOF torsional flutter is distinguishable from 2DOF flutter by the

fact that the A*

aerodynamic derivative (which is proportional to the aerodynamic

damping in torsion) changes sign from negative at low/fBto positive at some higher

value of/fB.

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

Cross-sections for which the A* aerodynamic derivative remains negative for allreduced wind speeds /fB(A*

negative"positive aerodynamic damping in torsion)

are likely to display 2DOF coupled vertical/torsional flutter behaviour. This occurs at

the wind speed where the motion-induced aerodynamic loads cause vertical and

torsional frequencies of motion to collapse into one common frequency. A mathemat-

ical model for coupled vertical/torsional flutter is presented by the following set of

one-degree-of-freedom oscillators assuming time complex harmonic vertical (h) and



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twisting () motion of the cross-section:

m[(!)#i2

]h"B

B

(iH*#H*)h

B#(iH*

#H*

),

(A.7)

I[(!)#i2]"BB

(iA*#A*)h

B#(iA*

#A*

),

(A.8)

where

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

2, A*

2 are the self-induced section loads the aerodynamic

derivatives.

It is noted that the aerodynamic loads introduces coupling between the equationsfor the vertical (h) and twisting () motion. Introducing the frequency ratio"/and the frequency ratio "/

and rearranging the equations of motion yields the

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

1!!B

mH*

#i2!

B

mH*

!

B

mH*

!

B

mH*

!B

I

A*!i

B

I

A* !!

B

I

A*#i

2!

B

I

A*

"0.

The flutter determinant defines a fourth-order real and a third-order imaginary

algebraic equation to be solved for introducing the H*2

, A*2

coefficients

obtained at successive values of the reduced wind speed /fB. Onset of 2DOF flutter

will occur at the particular reduced wind speed (/fB)where the roots of the real and

imaginary equations , are identical". Finally, the critical wind speed

for onset of 2DOF coupled flutter is obtained as

"

fB

fB. (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 IV, 194954.

[2] R.A. Frazer, C. Scruton, A summarised account of the severn bridge aerodynamic investigation, NPL

Aero Report, 222, London, HMSO, 1952.

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

233264.

[4] A. Larsen, Aerodynamic aspects of the final design of the 1624 m suspension bridge across the great

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