Comparison of Time Domain Techniques for the Evaluation of the Response and the Stability in Long...

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Computers and Structures 85 (2007) 1032–1048

Comparison of time domain techniques for the evaluation ofthe response and the stability in long span suspension bridges

Francesco Petrini a, Fabio Giuliano b,*, Franco Bontempi a

a Department of Civil Engineering, University ‘‘La Sapienza’’, Via Eudossiana 18, 00184 Rome, Italyb Department of Structural Mechanics, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy

Received 8 June 2006; accepted 20 November 2006Available online 16 January 2007

Abstract

During the last decades, several studies on suspension bridges under wind actions have been developed in civil engineering and manytechniques have been used to approach this structural problem both in time and frequency domain. In this paper, four types of timedomain techniques to evaluate the response and the stability of a long span suspension bridge are implemented: nonaeroelastic, steady,quasi steady, modified quasi steady. These techniques are compared considering both nonturbulent and turbulent flow wind modelling.The results show consistent differences both in the amplitude of the response and in the value of critical wind velocity.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Aeroelastic forces; Structural dynamics; Suspension bridge; Flutter; Buffeting

1. Introduction

Long span suspension bridges, because of their highflexibility, low structural damping, and the reduced massamount, are very sensitive to wind actions.

The actions of wind on a generic surface, are determinedby the configuration of punctual actions (pressure and tan-gential stresses), which are caused by the wind impact inthe relative motion.

For a flexible body inside a wind flow, these stresses aredetermined by the flow configuration in the surface prox-imity zone, and so depend on the body motion: from amechanic point of view, the couple of the flow and thebody, is an auto-excited dynamic system.

By discretizing the body to a finite number of degrees offreedom (DOFs), the equation governing the body motionis the dynamic equilibrium equation:

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doi:10.1016/j.compstruc.2006.11.015

* Corresponding author. Tel.: +39 328 5936901.E-mail address: [email protected] (F. Giuliano).

M � €qþ C � _qþ K � q ¼ F ðbody shape; q; _q; €q; V ; t; nÞ

ð1Þwhere

M mass matrix of the system,C damping matrix of the system,K stiffness matrix of the system,q; _q; €q DOFs of the system and their first an second time

derivates,V incident wind velocity,t time,n oscillation frequencies of the system.

In general, inside the right hand member of Eq. (1) thereis an ‘‘auto-excited’’ component of the aerodynamic forceswhich depends on body motion (q; _q; €q); in the case that theinertial terms, if compared with others terms, assume infin-itesimal order in Eq. (1), the equation becomes a staticequilibrium statement.

A global picture of all the aeroelastic problems that caninvolve a structure is represented by the ‘‘Collar triangle’’,as shown in the high-left side in Fig. 1: here, focusing on

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F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048 1033

aeroelastic forces dependences, aeroelastic problems areclassified in three categories:

1. Response problems, in which there is a dynamic equilib-rium between the body and the wind forces: the expres-sion of the forces contain time-independent terms.

2. Stability problems, in which the interchanging energybetween the body motion and the aeroelastic forces, pro-duces a gradual and unlimited increment of the motionenergy, leading to the dynamic equilibrium instability at‘‘critical velocities’’. For these problems, the expressionof the forces does not contain time-independent terms.

3. Mixed (stability and response) problems, in which inci-dent wind velocities are close to the critical velocitiesand the expression of the forces contains both time-inde-pendent and auto-excited terms.

In instability problems, the forces are usually transferredin the left side hand of the equation in order to obtain ahomogeneous equation in which mass, damping, and stiff-ness matrices contain auto-excited coefficients.

Considering a generic structure under wind action, twoprincipal approaches exist to solve the relative aeroelasticproblem:

1. Frequency domain approach, usual for instability prob-lems, based on a modal combinations for the structuralsystem, is less reliable for structures with strong nonlin-ear nature.

Aeroelastic problems

For stability

For Response

An asymmetry in the flowproduces a vertical oscillationthat generate oscillations inaerodynamic forces, which depends on body oscillationsvelocity and involves anaerodynamic damping, which is opposite to the structural damping. If aerodynamic damping is greater than thestructural one, the motion may become unstable.

Galloping

Static instability in which torsional moment due toaeroelastic forces, overcomes the elastic resistant moment ofthe body

Torsional divergence

Dynamic instabilityto a resonance. Sloose themselves behind the body. configuration geoscillation in forces, with frequency. In a cerwind velocity, thfrequency of the fothe structural frequency and it structural oscillatio

Vortex She

Aeroelastic PhenomenaTors. DivergenceV. SheddingGallopingClassical flutterStall flutterBuffeting

DVSGFcFsB

Forces: A=Aeroelastic; E=Elastic; I=Inertial; F=Applied

Collar

Non Aeroelastic Sciences

Mechanics of MVvibrations

Mechanics of MRrigid body flight

Equilibrium equation: A + E + I + F = 0

Aeroelastic problems

For stability

For Response

An asymmetry in the flowproduces a vertical oscillationthat generate oscillations inaerodynamic forces, which depends on body oscillationsvelocity and involves anaerodynamic damping, which is opposite to the structural damping. If aerodynamic damping is greater than thestructural one, the motion may become unstable.

Galloping

Static instability in which torsional moment due toaeroelastic forces, overcomes the elastic resistant moment ofthe body


Dynamic instabilityto a resonance. Sloose themselves behind the body. configuration geoscillation in forces, with frequency. In a cerwind velocity, thfrequency of the fothe structural frequency and it structural oscillatio

Vortex She

Aeroelastic PhenomenaTors. DivergenceV. SheddingGallopingClassical flutterStall flutterBuffeting

DVSGFcFsB

Forces: A=Aeroelastic; E=Elastic; I=Inertial; F=Applied

Collar

Non Aeroelastic Sciences

Mechanics of MVvibrations

Mechanics of MRrigid body flight

Equilibrium equation: A + E + I + F = 0

Fig. 1. Aeroelastic probl

2. Time domain approach, requiring a time integration ofthe equations of the motions of the structure subjectedto wind loads, with large computational costs for struc-ture with many DOFs.

In the present paper, the time domain approach hasbeen adopted to study the response and the stability prob-lem of a long span suspension bridge under wind actions.In particular, the incident wind has been modelled as a tur-bulent flow for the response problem, while for the stabilityproblem, it has been modelled as nonturbulent.

2. Aeroelastic forces for suspension bridges

In the last decades, referring to aeronautic engineering,civil engineers have developed many analytical theories tomodel wind effects on structures, and simplified approacheshave been adopted to evaluate the aeroelastic terms,because of the very complex dependence above listed. Bothfrequency and time domain techniques to model aeroelasticforces, derive from wing theory [1–3,5].

In the case of suspension bridges, two peculiar aspectshave to be considered. First, suspension bridges deck sec-tions, usually have so-called ‘‘semi-bluff’’ or ‘‘bluff’’ bodyshapes, with (opposite to wing shapes) well predictablepoints of flow detachments along the deck surfaces (‘‘liveedges’’), where the turbulence of the flow is very high.The second particular aspect is related to the intrinsicturbulent content of an incident wind flow. Furthermore

Static aeroelastic stability

Dynamic aeroelastic stability

Static aeroelastic response

Dynamic aeroelastic response

0=− EA

0=−+ EFA

0=−+ EIA

0=+−+ FEIA


Galloping

Flutter

Buffeting

Vortex Shedding

Dynamic instability in which two d.o.f. of the forcedstructural system are coupledand, under opportune configurations (defined “critic”) for frequencies and reciprocalphase angles, lead the damping of the system to becomenegative, and the oscillationsincrease in amplitude

Flutter

very similar ome vortex

in the wakeSuch physic nerates anaerodynamic a definite tain range of e oscillation rces “block” oscillations

governs the n period.

dding

Motion under the action of timerandom variable forces both inintensity and direction. The buffeting phenomenon assumesaeroelastic importance in concomitance of other aeroelastic phenomena (like Flutter or Vortex shedding).

A classic buffeting effect, is the one generated by the intrinsic turbulence within the atmospheric boundary layer.

Buffeting

VS

D

G

F

B

Static aeroelastic stability

Dynamic aeroelastic stability

Static aeroelastic response

Dynamic aeroelastic response

0=− EA 0=− EA

0=−+ EFA 0=−+ EFA

0=−+ EIA 0=−+ EIA

0=+−+ FEIA 0=+−+ FEIA


Galloping

Flutter

Buffeting

Vortex Shedding

Dynamic instability in which two d.o.f. of the forcedstructural system are coupledand, under opportune configurations (defined “critic”) for frequencies and reciprocalphase angles, lead the damping of the system to becomenegative, and the oscillationsincrease in amplitude

Flutter

very similar ome vortex

in the wakeSuch physic nerates anaerodynamic a definite tain range of e oscillation rces “block” oscillations

governs the n period.

dding

Motion under the action of timerandom variable forces both inintensity and direction. The buffeting phenomenon assumesaeroelastic importance in concomitance of other aeroelastic phenomena (like Flutter or Vortex shedding).

A classic buffeting effect, is the one generated by the intrinsic turbulence within the atmospheric boundary layer.

Buffeting

VS

D

G

F

B

ems and instabilities.

1034 F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048

a larger complexity and rotationally nature of the flowdetermines a larger ‘‘aerodynamic delay’’, which is thetransient effect due to the adjustment time of aerodynamicfield, in consequence of a changing in body geometric con-figuration (i.e. rotation and displacements of the deck). Inorder to take into account these effects, the so-called‘‘memory terms’’, which consider the influence of displace-ments history in the expression of the aeroelastic forces, areintroduced. These terms are usual implemented by integralexpressions [6].

The uncertainties related to the above phenomena leadto the use of the experimental approach. In this sense, stud-ies conduced by Scanlan and Tomko [7], Jain et al. [8] andScanlan and Simiu [9] had great relevance in wind engineer-ing. Following Scanlan theory, the Self-Excited (SE) com-ponents of the aeroelastic forces are determined by asuperposition of the effects, referring to the forces obtainedby wind tunnel tests, acting on a sectional model of thedeck that is moving in simple harmonic oscillations alongthree sectional DOFs (the rotational and the two transla-tional ones). Referring to Fig. 2, it results for the Lift force

LSEðp; _p; h; _h; #; _#; k;xÞ

¼ 1

2� q � V 2 � B � k � H �1ðkÞ �

_hVþ k � H �2ðkÞ �

B � _#

V

"

þ k2 � H �3ðkÞ � #þ k2 � H �4 �hBþ k � H �5ðkÞ �

_pV

þ k2 � H �6ðkÞ �pB

#ð2Þ

where

k ¼ x�BV reduced frequency of the system, in simple har-

monic motion,H�i ðkÞ functions of the reduced frequency,B characteristic dimension of the bridge deck,x circular frequency of the system, in simple har-

monic motion,V incident wind velocity.

The functions H �i ðkÞ are called ‘‘flutter derivatives’’ ofthe bridge deck, and are determined by wind tunnel tests,imposing simple harmonic motion to the deck model.

L(t1)M(t1)

D(t1)

p h

B

x

yV

ϑ

L(t2) M(t2)

D(t2)

L(t1)M(t1)

D(t1)

p h

B

x

yV

ϑ

L(t2) M(t2)

D(t2)

Fig. 2. Bridge deck section under wind action.

Analogous expressions of (2) can be written for dragforce and moment.

3. Aeroelastic forces in time domain

Time domain approaches allow to consider directly thestructural nonlinear effects and this is relevant for certaintypes of structures, like long span suspension bridges. Fur-thermore, because the time domain analyses outputs aremerely the time histories of specific variables, it is the mostconvenient approach in response problems. Unfortunately,expressions in time domain which consider aspectspreviously mentioned are not trivial to implement. Conse-quently in the last years simplified formulations for aero-elastic forces have been developed and improved.

The analysis in the time domain consists in a time inte-gration that involves the time step updating of kinematicparameters and acting forces. Referring to Fig. 3, wherethe problem is represented like a two-dimensional problem,horizontal and vertical components of absolute wind tur-bulent velocity V aðtÞ, are considered as composed by meancomponents U, W, and fluctuant (or turbulent) compo-nents uðtÞ and wðtÞ. The resulting absolute velocity is nothorizontal, and has a time-varying instantaneous angle ofincidence.

Adopting the general notation previously introduced(Eq. (1)), one has, for the system DOFs, qT ¼ ½ s h # �.Moreover, the dependence of the forces from structuralDOFs and their time derivatives can be generally expressedin matrix form as:

F ðq; _q; €q; nÞ ¼ Pðt; nÞ � €qþ Qðt; nÞ � _qþ Rðt; nÞ � q ð3Þ

where the time dependence of DOFs has not been noticed:Pðt; nÞ;Qðt; nÞ and Rðt; nÞ represent in expression (3) coeffi-cient matrices which, in the most general case, depends ontime and on oscillation frequency of the system.

Principal characteristics of the various aeroelastic theo-ries, are summarized in Table 1: it is clear the overall com-plexity of the classification, while the most used timedomain techniques are presented below.

L(t2)M(t2)

D(t2)

p

h

B

x

y

Va(t2)

ϑ

Vx=U+u

Vy=W+w

αL(t1)

M(t1)

D(t1)

L(t2)M(t2)

D(t2)

p

h

B

x

y

Va(t2)

ϑ

Vx=U+u

Vy=W+w

αL(t1)

M(t1)

D(t1)

Fig. 3. Bridge deck section under turbulent wind action.

Table 1Aeroelastic theories

Level Theory Hypothesis and approximations

General form F ðq; _q; €q; t; nÞ(Ref. [9])

DOFsdependences

Dependentfrom n

Nonlinearisedpolar lines

S.E.P.a Aerodynamiccoefficients

qðtÞ _qðtÞ €qðtÞ0 Nonaeroelastic (NO) R X Static1 Steady (ST) R � qðtÞ X X Static2 Quasi Steady (QS) R � qðtÞ þ Q� _qðtÞ X X X Static3 Modified Quasi Steady

(QSM)RðtÞ � qðtÞ þ QðtÞ � _qðtÞ X X X X Dynamic

4 Extension of AeroelasticDerivates in Time Domain(ADTD)

Pðt; nÞ � €qþ Qðt; nÞ � _qþ Rðt; nÞ � q X X X X X X Dynamic

4 Scanlan Theory (NSS) Frequency domain X X X X X Dynamic

a S.E.P. = Superposition of Effects Principle (concerning the determination of the actions).


3.1. Nonaeroelastic theory (NO)

This is a ‘‘zero level’’ aeroelastic theory: aeroelasticeffects are not considered in the forces formulation butonly the relative angle of incidence between wind and deck,change with time just in accordance with the turbulence ofthe incident wind. Adopting the small displacementshypothesis, (linearised lift and moment polar line), itresults

DðtÞ ¼ 1

2q � jV aðtÞj2 � B � cD½aðtÞ�

LðtÞ ¼ 1

2q � jV aðtÞj2 � B � KL0 � aðtÞ

MðtÞ ¼ 1

2q � jV aðtÞj2 � B2 � KM0 � aðtÞ

ð4Þ

where a is the angle of attack, cD is drag coefficient andKL0, KM0 are the angular coefficients of lift and momentpolar diagrams, respectively.

3.2. Steady theory (ST)

This is a ‘‘first level’’ aeroelastic theory, where the rela-tive angle of incidence between wind and deck, changeswith time due to both the incident wind turbulence andthe rotation (torsion) of the deck. Supposing that thebridge deck section rotates around a mean equilibriumposition # ¼ #0, adopting the small displacements hypoth-esis (both lift and moment polar diagrams are linearised),the aerodynamic coefficients become

cLðcÞ ¼ cLð#0Þ þ KL0 � ðc� #0ÞcMðcÞ ¼ cMð#0Þ þ KM0 � ðc� #0Þ

ð5Þ

in which KL0 and KM0 are the angular coefficients of polarlines computed in # ¼ #0. Referring to Fig. 3 and definingcðtÞ ¼ aðtÞ � #ðtÞ, the aeroelastic forces are expressed as

DðtÞ ¼ 1

2q � jV aðtÞj2 � B � cD½cðtÞ�

LðtÞ ¼ 1

2q � jV aðtÞj2 � B � cL½cðtÞ�

MðtÞ ¼ 1

2q � jV aðtÞj2 � B2 � cM½cðtÞ�

ð6Þ

Adopting the general formulation of Eq. (3), one can write

F ¼ F ðq; tÞ ¼ R � qðtÞ ð7Þ

The steady theory has the appeal of simplicity; further-more, in the case of nonturbulent incident wind, it showsthe fundamental mechanisms of a flutter stability problem(coalescence of frequency, influence of structural parame-ters, etc.). Nevertheless it implies many approximations,such as the neglecting of the dependence of aeroelasticforces on structural velocities, accelerations and oscillationfrequency, and the linearisation of the relation betweenaeroelastic forces and structural DOFs. Furthermore, thesteady theory does not consider the aerodynamic delay,and utilizes static aerodynamic coefficients.

3.3. Quasi Steady theory (QS)

It is a ‘‘second level’’ aeroelastic theory: instantaneousaeroelastic forces acting on the structure are the same thatact on the structure itself when it moves with constanttranslational and rotational velocities, equal to the realinstantaneous ones. The main assumption consists in con-sidering that the body (deck section) is motionless, togetherwith the wind having velocities and directions equal to theinstantaneous relative (wind-deck) ones: such assumptionis represented in Fig. 4.

The coefficients of _# (bi, with i ¼ L;M) should be derivedexperimentally by wind tunnel tests [10]; it can be derivedalso through the use of Computational Fluid Dynamic(CFD) techniques [11,12].

Adopting the hypothesis of small displacements aroundthe mean configuration, Eq. (5) are also valid and the

L(t2)M(t2)

D(t2)

p

h

B

x

y

Va(t2)

ϑ

Vx=U+u

Vy=W+w

α

ϑhh ⋅⋅+− Bbh i

β

ph−

L(t1)M(t1)

D(t1)

L(t2)M(t2)

D(t2)

p

h

B

x

y

Va(t2)

ϑ

Vx=U+u

Vy=W+w

α

ϑ·· ⋅⋅+− Bbh i

β

p·−

L(t1)M(t1)

D(t1)

Fig. 4. Quasi steady theory assumption.


expressions of aeroelastic forces are identical (in the form)

to the steady theory ones, with biðtÞ ¼ arctg V y ðtÞ� _hþbi �B� _#ðtÞV x� _p

� �(i ¼ L;M) in substitution of aðtÞ:

DðtÞ ¼ 1

2q � jV aLðtÞj2 � B � cD½cðtÞ�

LðtÞ ¼ 1

2q � jV aLðtÞj2 � B � cL½cðtÞ�

MðtÞ ¼ 1

2q � jV aMðtÞj2 � B2 � cM½cðtÞ�

ð8Þ

where ciðtÞ ¼ biðtÞ � #ðtÞ and jV aiðtÞj2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðV x � _pÞ2 þ ðV y � _hþ bi � B � _#Þ2

q. Adopting the general

formulation of Eq. (3), one can write

F ¼ F ðq; _q; tÞ ¼ R � qðtÞ þ Q � _qðtÞ ð9Þ

The Quasi Steady theory can consider the dependences ofaeroelastic forces from structural velocities, preserving alsoa relatively simple algorithmic implementation. Further-more, the dependence from oscillation frequency is ne-glected, and the dependence of aeroelastic forces from theDOFs of the structure is linearised. The Quasi Steady the-ory does not consider the aerodynamic delay, utilizing sta-tic aerodynamic coefficients, with the possible exception forthe bi coefficients (with i ¼ L;M), whose value are dynam-ically assessed [10].

Considering the expected low incidence of turbulentcomponent on the auto-excited forces, and neglecting thehigh order infinitesimal terms, it is possible to obtain [10]more elegant and explicit expressions than (8), in which sta-tic, auto-excited and buffeting component are outlined andexpressed separately.

3.4. Modified Quasi Steady theory (QSM)

In this ‘‘third level’’ aeroelastic theory, in respect to theQS theory, the only changes concern the aerodynamic coef-ficients for the lift and the moment, which become dynamicas measured by wind tunnel tests [13]. Referring to Fig. 4,aeroelastic forces are expressed by the followingexpressions:

DðtÞ ¼ 1

2q � jV aLðtÞj2 � B � cD½cðtÞ�

LðtÞ ¼ 1

2q � jV aLðtÞj2 � B � c�L½cðtÞ�

MðtÞ ¼ 1

2q � jV aMðtÞj2 � B2 � c�M½cðtÞ�

ð10Þ

where ciðtÞ, jV aiðtÞj2 (i ¼ L;MÞ and cD, have the samemeaning as the previous expressions included in QS theory.In the expressions (10), aerodynamic coefficients c�L and c�Mare dynamic and they are computed like below

c�L ¼ cLð#0Þ þZ #

#0

KL d �#

c�M ¼ cMð#0Þ þZ #

#0

KM d �#

ð11Þ

where cLð#0Þ and cMð#0Þ are the static aerodynamic coeffi-cients computed in the mean equilibrium configuration(# ¼ #0), and KL, KM are the ‘‘dynamic derivatives’’ com-puted like below

KL ¼ h3 �ocL

o#

� �#¼ �#

KM ¼ a3 �ocM

o#

� �#¼ �#

ð12Þ

where h3 and a3 are the Zasso’s theory coefficients [15], as-sessed by dynamic wind tunnel tests. These coefficients aresimilar to the Scanlan’s motion derivatives (2), and they de-pend both from the rotation deck angle and the ‘‘reducedwind velocity’’ V red ¼ V =ðx � BÞ (depending from x, whichis the motion frequency). For multi-degree of freedomstructures (MDOFs), the motion frequency is a combina-tion of overall mode shape frequencies, and for nonlinearstructures it varies at every instant, depending on the stateof structure. So the in advance computation of h3 and a3 isnot practicable. To overcome this problem, in the QSMtheory, the fundamental frequency of the structure is usedto compute the reduced velocity V red ¼ V =ðx � BÞ and thecorresponding h3 and a3 coefficients. Therefore, the depen-dence of aeroelastic forces from the motion frequency isnot considered.

Adopting the general formulation, one can write

F ¼ F ðq; _q; tÞ ¼ RðtÞ � qðtÞ þ QðtÞ � _qðtÞ ð13Þ

The QSM theory has the attractive aspects of the QS the-ory (together with high analytic difficulty), and implementsdynamic aerodynamic coefficients. Such coefficients takeinto account the nonlinearity of the response in respect tothe wind angle of attack, taking also a partial considerationof the aerodynamic delay. Furthermore they do not con-sider the dependence of the forces from the oscillationmotion frequency.


3.5. Theory of aeroelastic derivates in time domain (ADTD)

In this ‘‘fourth level’’ aeroelastic theory, the basic con-cept is very similar to the Wagner’s indicial function theory[6]. The auto-excited component of aeroelastic forces iscomputed by a convolution integral:

DSEðtÞ ¼Z t

�1ðIDSEhðt � sÞ � hðsÞ

þ IDSEpðt � sÞ � pðsÞ þ IDSE#ðt � sÞ � #ðsÞ � ds

LSEðtÞ ¼Z t

�1ðILSEhðt � sÞ � hðsÞ

þ ILSEpðt � sÞ � pðsÞ þ ILSE#ðt � sÞ � #ðsÞÞ � ds

MSEðtÞ ¼Z t

�1ðIMSEhðt � sÞ � hðsÞ

þ IMSEpðt � sÞ � pðsÞ þ IMSE#ðt � sÞ � #ðsÞÞ � ds

ð14Þ

where I iSEj (i ¼ D; L; M and j ¼ p; h; #) is the impulsivefunction of the auto-excited force i which correspondsto the generic jth DOF. Such function represents the aero-elastic force component i which acts on a body under awind flow which has an impulsive motion along the jthDOF. By a Fourier transformation of Eq. (14), and sup-posing that the motions along the three DOFs are sinusoi-dal with the same oscillation frequency, by comparing Eq.(14) with Eq. (2), it is possible to obtain the relationshipsbetween the Fourier transform (I iSEj ) of I iSEj (i ¼ D; L;Mand j ¼ p; h; #), and the Scanlan’s flutter derivatives: ifone knows the Scanlan’s flutter derivatives of lift, dragand moment, one can obtain also the functions I iSEj [6].Nevertheless the flutter derivatives are known only indiscrete values of reduced frequency (k), and are madecontinuous in the frequency domain by the Roger’sapproximating function [6] that can replace the I iSEj . Oper-ating a changing of variable, the Roger’s function aretransposed in Laplace’s domain and, by the Laplace’s in-verse transformation, they are transposed finally in thetime domain. Concerning, for example, the part of auto-excited component of the Lift that depends on the sec-tional vertical DOF (h(t)), using this procedure one canobtain the following expression:

LSEhðtÞ ¼1

2q � jV aðtÞj2 � B

� a1 � hðtÞ þ a2 �BjV aj� _hþ a3 �

B2

jV aj2� €hþ

Xm

l¼1

/lðtÞ !

ð15Þ

where ai coefficients and the sum extreme m are those pre-viously defined (during the Roger’s function generationphase), and the /lðtÞ are integral terms, which representthe ‘‘memory terms’’ of the force. The assessment of /lðtÞ

functions needs the introduction of further m differentialequations in the /lðtÞ and h(t) functions [6].

Adopting the general formulation one can write

F ðq; _q; €q; nÞ ¼ P ðt; nÞ � €qþ Qðt; nÞ � _qþ Rðt; nÞ � q ð16Þ

From a conceptual point of view, such theory is the mostcomplete among the time domain formulations: the depen-dences of aeroelastic forces both on the structural DOFsand on structural velocities and accelerations are imple-mented, and that on the motion frequency is also consid-ered. Furthermore the aerodynamic delay is quantified byRoger’s formulas. Otherwise one can note a great increasein the analytical difficulties of the method in respect to theothers.

4. Application on a long span suspension bridge

In this paper, using the above introduced time domaintechniques, the response problem of a long span suspensionbridge under turbulent wind, has been studied. After that,the stability problem under nonturbulent wind has beenstudied, comparing the critical velocities computed by dif-ferent techniques.

4.1. Descriptions of the bridge and structural performanceaspects

A long span suspension bridge has been examined [16].The main span of the bridge is 3300 m long, while the totallength of the deck, 60 m wide, is 3666 m (including the sidespans). The deck is formed by three box sections, the outerones for the roadways and the central one for the railway(Fig. 6). The roadway deck has three lanes for each car-riageway (two driving lanes and one emergency lane), each3.75 m wide, while the railway section has two tracks.

The two towers are 383 m high and the bridge suspen-sion system relies on two pairs of steel cables, North andSouth, each with a diameter of 1.24 m and a total length,between the anchor blocks, of approximately 5000 m. Prin-cipal characteristics of the structure are summarized inFigs. 5 and 6. Because of the suspended span size, the sen-sitivity of the structure at the wind action is foreseeable. Inthis sense, the adopted ‘‘multibox’’ section [17] for the decksection is innovative and it is finalized to optimize the aero-dynamic response.

The numerical model was based on the preliminarydesign of the Messina Strait Bridge (for a complete descrip-tion of geometrical and mechanical properties, see [18]) andit has been developed using 3D beam finite elements, witheach node having six degrees of freedom, as shown inFig. 7. The permanent loads and the masses are modelledas distributed along the elements. For the developed tran-sient step by step analyses, a Newmark time integrationscheme [19,20] has been adopted, in which geometric non-linearities has been considered.

3300183 183777 627960 3300 m 810

+77.00 m

+383.00 +383.00

+54.00+118.00

+52.00 +63.00

Fig. 5. Bridge profile.

Fig. 6. Bridge deck section.

Fig. 7. 3D FEM model.


In the design stages, numerical analyses are conducted inorder to verify safety and serviceability performance of thebridge, organized in contingency scenarios and related todifferent probabilities of occurrence, i.e., deck accelera-tions, stresses on substructures, critical wind velocities.Expected values of each performance are fixed in the basisof design and verified by structural analyses [21].

4.2. Analyses developed

The structural response has been investigated in respectto three different mean wind velocities at the deck level: 21,45, 57 m/s, which correspond to a 50, 200 and 2000 years ofreturn period TR, and to three different structural require-

ments: complete serviceability (roadway and railwaytraffic), partial (only railway traffic) serviceability andmaintaining the structural integrity respectively. Nonaero-elastic (NO), steady (ST), quasi steady (QS) and modifiedquasi steady (QSM) theories have been applied in the anal-yses. Both nonturbulent and turbulent flows have beenconsidered. The results of the analyses are listed in Tables2 and 3.

4.3. Results

4.3.1. Response problemSeveral geometric configurations of the bridge deck

section (obtained by changing the shape, the traffic and

Table 2Response analyses

Wind meanvelocity (m/s)

Nonturbulentflow

Turbulentflow

Response analyses done by aeroelastic theories: NO, ST, QS, QSM

Time history of midspandisplacements

45 Fig. 9 Figs. 12–15

Statistics 45 Figs. 12–15

Envelopes of deckdisplacements

21 Fig. 1645 Fig. 11 Fig. 1657 Fig. 16

Envelopes of deckvelocities andacceleration

45 Fig. 17

Table 3Stability analyses

Formulation Nonturbulentflow

Stability analyses done by aeroelastic theories

Time history of midspandisplacements

ALL Fig. 20

Critical velocities ALL Fig. 21Diagram on phases plane QS Fig. 23Aerodynamic damping on wind

velocitiesQS Fig. 24

L

D

M. a-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-10 -8 -6 -4 -2 0 2 4 6 8 10

[deg]

Drag Lift Moment

L

D

M, a-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-10 -8 -6 -4 -2 0 2 4 6 8 10

[deg]

Drag Lift Moment

Fig. 8. Polar lines for response problem.


wind barriers configurations) have been tested in wind tun-nel tests: in Fig. 8, the static polar lines used in this paperfor the response problem are reported.

At first, a preliminary analysis of the responses undernonturbulent wind has been performed to assess the funda-mental characteristics of the responses (oscillation ampli-tudes, aerodynamic damping, mean values), supplied bythe different theories, regardless the dispersion of resultsinduced by the turbulence. In Fig. 9, oscillations alongthe three sectional DOFs of the railway box mass centrein the deck midspan, computed by an incident nonturbu-lent flow having a mean velocity of 45 m/s, are shown. InFig. 10 the relative mean values are presented, togetherwith the experimental ones [23].

The response of the structure is represented by a timedamped oscillation. In QS and QSM results, one can notethe presence of an aerodynamic damping (QðtÞ � _qðtÞ, with

reference to the general form), so that the oscillation ampli-tude decreases more than linearly in time. Concerning themean values, they result quite greater than the experimen-tal ones, especially for the deck rotation.

In Fig. 11, time envelopes of transversal and verticaldeck displacements (railway box section mass centre) undernonturbulent flow having a mean velocity of 45 m/s arepresented, together with the results derived from a staticequivalent formulation, and static analysis. One can note

that the relative differences from a theory to another arenot significant.

After preliminary nonturbulent flow analyses, successiveanalyses have been conducted considering a turbulentwind. Time histories of the wind velocity field have beengenerated numerically and obtained by Solari and Caras-sale [22], and are generated like components of a multivar-iate, multidimensional Gaussian stationary stochasticprocess.

In Figs. 12–14, oscillations along the three sectionalDOFs of the railway box mass centre in the deck midspan,computed by an incident turbulent flow having a meanvelocity of 45 m/s, are represented. Every displacementtime history has been characterized from a statistic pointof view by the frequency probability density (includingthe 5% and 95% fractile values), and by the histogram rep-resenting the overcoming frequencies.

In Fig. 15, time histories for the three sectional DOFs ofthe railway box mass centre in the deck midspan, computedby an incident turbulent flow having a mean velocity of45 m/s, are resumed, and also the computed and experi-mental mean values are represented.

In general, one can note that by increasing the complex-ity of the aeroelastic forces representation (followingthe succession NO, ST, QS, QSM), both the maximumamplitude of the oscillations and the variance of computedtime history decrease. Regarding this tendency an excep-tion is represented from the rotation of the deck aroundown longitudinal axis, which in QSM results is greater(both in amplitude and in dispersion) than that obtainedby QS.

One can note that NO results are a similar to those ofST results, and QS results are similar to the QSM ones.Concerning the mean values, the similitude of the resultsfor couples of formulations (NO–ST, QS–QSM) isconfirmed.

Concerning to the mean incident wind velocities of 21and 57 m/s, analogous analyses have been conducted. InFig. 16, regarding the three examined velocities, time enve-lopes of the transversal and vertical deck displacementsunder turbulent flow are represented.

Transversal Vertical Rotation

NO

5.40

5.45

5.50

5.55

5.60

5.65

400 900 1400 1900 2400 2900

time (sec)

Uy

(m)

-0.295

-0.291

-0.287

-0.283

-0.279

-0.275

400 900 1400 1900 2400 2900

time (sec)

Uz

(m)

-0.0091

-0.0089

-0.0087

-0.0085

-0.0083

-0.0081

400 900 1400 1900 2400 2900

time (sec)

Rot

(R

AD

)

ST

5.40

5.45

5.50

5.55

5.60

5.65

400 900 1400 1900 2400 2900

time (sec)

Uy

(m)

-0.245

-0.241

-0.237

-0.233

-0.229

-0.225

400 900 1400 1900 2400 2900

time (sec)

Uz

(m)

-0.0098

-0.0096

-0.0094

-0.0092

-0.0090

-0.0088

400 900 1400 1900 2400 2900

time (sec)

Rot

(R

AD

)

QS

400 900 1400 1900 2400 2900

ti me (sec)

Uy

(m)

-0.358

-0.354

-0.350

-0.346

-0.342

-0.338

400 900 1400 1900 2400 2900

time (sec)

Uz

(m)

-0.0084

-0.0082

-0.0080

-0.0078

-0.0076

-0.0074

400 900 1400 1900 2400 2900

time (sec)

Rot

(R

AD

)

QS

M

5.40

5.45

5.50

5.55

5.60

5.65

400 900 1400 1900 2400 2900

ti me (sec )

Uy

(m)

-0.348

-0.344

-0.340

-0.336

-0.332

-0.328

400 900 1400 1900 2400 2900

time (sec)

Uz

(m)

-0.0084

-0.0082

-0.0080

-0.0078

-0.0076

-0.0074

400 900 1400 1900 2400 2900

time (sec)

Ro

t (R

AD

)

5.40

5.45

5.50

5.55

5.60

5.65

Fig. 9. Time history of midspan displacements (nonturbulent flow; V ¼ 45 m/s).

Transversal Max Vertical Max

NOST QS QSM

Static

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 1000 2000 3000 4000

abscissa (m)

Uz

(m)

Static

NO

ST

QS

QSM

-0.4

-0.3

-0.2

-0.1

0.0

0 1000 2000 3000 4000

abscissa (m)

Uz

(m)

Fig. 11. Envelopes of deck displacements (nonturbulent flow; V ¼ 45 m=s).

Transversal Vertical Rotation

NO ST QS QSM Experim-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

Ro

tati

on

(DE

G)

Ro

tati

on

(DE

G)

NO ST QS QSM Experim NO ST QS QSM Experim

0.0

-0.1

-0.2

-0.3

-0.4

6.0

5.0

4.0

3.0

2.0

1.0

0.0

Fig. 10. Mean values of midspan displacements (nonturbulent flow; V ¼ 45 m/s).


Time history Frequencies Probability density

NO

0

2

4

6

8

10

12

14

400 900 1400 1900 2400 2900

time (sec)

Uy

(m)

0

200

400

600

800

1000

1200

2,41 3,91 5,42 6,92 8,43 9,9311,4

312,9

4

Class

Fre

qu

ency

ST

0

2

4

6

8

10

12

14

400 900 1400 1900 2400 2900

time (sec)

Uy

(m)

0

200

400

600

800

1000

1200

2,41

3,91

5,42

6,92

8,43

9,93

11,4

312

,94

Class

Fre

qu

ency

QS

0

2

4

6

8

10

12

14

400 900 1400 1900 2400 2900

time (sec)

Uy

(m)

0

200

400

600

800

1000

1200

2,41

3,91

5,42

6,92

8,43

9,93

11,4

312

,94

Class

Fre

qu

ency

QS

M

0

2

4

6

8

10

12

14

400 900 1400 1900 2400 2900

time (sec)

Uy

(m)

0

200

400

600

800

1000

1200

2,41

3,91

5,42

6,92

8,43

9,93

11,4

312

,94

Class

Fre

qu

ency

Fig. 12. Time histories of midspan transversal displacements and their statistic characterization (turbulent flow; V ¼ 45 m=s).


Envelopes confirm the tendency previous evidenced bythe time histories of midspan displacements: increasingthe complexity of the aeroelastic forces representation,the envelopes decrease. Also in terms of envelopes, thesimilitude of the results for couples of formulations (NO–ST, QS–QSM) is confirmed. Among the examined formu-lations, the ST is the more sensitive to the increase of meanwind velocity.

Similar diagrams have been computed concerning veloc-ities and accelerations of the deck: in Fig. 17 the time enve-lopes of these kinematic entities are represented for thewind mean velocity of 45 m/s. The deck accelerations, inparticular, have a great relevance in the bridge perfor-mance table: they have to maintain themselves under animposed limit to ensure the safety during the transit ofthe trains.

Also in the velocities and in the accelerations envelopes,there is a decrease when the complexities of the formula-tions increase, and the results are similar by couple offormulations (NO–ST, QS–QSM).

4.3.2. Stability problemOnce evaluated the response under turbulent incident

flow, further analyses have been conducted for the aero-elastic stability problem under nonturbulent wind. Typi-cally, for suspension bridges, the most dangerousinstability phenomenon is flutter (Fig. 1), a dynamic insta-bility in which two DOFs of the forced structural systemare coupled: under opportune configurations (defined‘‘critical’’) for frequencies and reciprocal phase angles, itmakes the damping of the system become negative, andthe structural oscillations increase in amplitude. For a


NO

-3.5

-2.5

-1.5

-0.5

0.5

1.5

2.5

3.5

400 900 1400 1900 2400 2900

time (sec)

Uz

(m)

0

200

400

600

800

1000

1200

-1.7

9-1

.02

-0.2

50.

531.

302.

072.

843.

61

Class

Fre

qu

ency

ST

-3.5

-2.5

-1.5

-0.5

0.5

1.5

2.5

3.5

400 900 1400 1900 2400 2900

time (sec)

Uz

(m)

0

200

400

600

800

1000

1200

-1.7

9-1

.02

-0.2

50.

531.

302.

072.

843.

61

Class

Fre

qu

ency

QS

-3.5

-2.5

-1.5

-0.5

0.5

1.5

2.5

3.5

400 900 1400 1900 2400 2900

time (sec)

Uz

(m)

0

200

400

600

800

1000

1200

1400

1600

1800

-1.79

-1.02

-0.25

0.53 1.30 2.07 2.84 3.61

Class

Fre

qu

ency

QS

M

-3.5

-2.5

-1.5

-0.5

0.5

1.5

2.5

3.5

400 900 1400 1900 2400 2900

time (sec)

Uz

(m)

0

500

1000

1500

2000

2500

3000

3500

-1.79

-1.02

-0.25

0.53 1.30 2.07 2.84 3.61

Class

Fre

qu

ency

Fig. 13. Time histories of midspan vertical displacements and their statistic characterization (turbulent flow; V ¼ 45 m=s).


suspension bridge deck, the two above sectional DOFs, arethe vertical and the rotational one. Critical configurationsare such as that the oscillation frequency is the same andthe difference between the phase angles is equal to p/2.Samples of stable, critical and unstable oscillations areshown in Fig. 18.

The vertical and, in particular, rotational motion fre-quency, depend on the incident wind velocity. When thisvelocity increases, the frequencies come closer to each otheruntil the ‘‘frequency coalescence’’: during this interval oftime the damping is positive. When the two frequenciescoincide, the damping becomes equal to zero and, if thewind velocity increases, the damping of the system becomesnegative. The wind velocity which corresponds to zerodamping and incipient flutter is called ‘‘critical wind veloc-ity’’ (Vcrit).

The capability of the examined formulations in comput-ing the flutter phenomenon has been investigated. InFig. 19, the polar lines that have been utilized in the stabil-ity problem are shown.

Concerning the NO formulation, it is evident from Eq.(4) that the forces do not depend on the structure motion,in the case of nonturbulent incident wind, and the forcesare constant in time. An increase of the wind velocity pro-duces an increase of initial amplitudes of the structure oscil-lation only, so oscillations decrease in time. Consequently,the NO theory is unable to compute the flutter, while theothers formulations can represent the phenomenon, butthey lead to different values of the critical wind velocity.

In Fig. 20, time histories of unstable oscillations(V > V crit) are shown. The diagrams refer to the ST, QSand QSM theories.


NO

-0.055

-0.045

-0.035

-0.025

-0.015

-0.005

0.005

0.015

0.025

400 900 1400 1900 2400 2900

time (sec)

Rot

(R

AD

)

0

200

400

600

800

1000

1200

-0.047

-0.036

-0.025

-0.014

-0.003

0.008

0.019

0.030

Class

Fre

qu

ency

ST

-0.055

-0.045

-0.035

-0.025

-0.015

-0.005

0.005

0.015

0.025

400 900 1400 1900 2400 2900

time (sec)

Rot

(R

AD

)

0

200

400

600

800

1000

1200

-0.0

47

-0.0

36

-0.0

25

-0.0

14

-0.0

030.

008

0.01

90.

030

Class

Fre

qu

ency

QS

-0.055

-0.045

-0.035

-0.025

-0.015

-0.005

0.005

0.015

0.025

400 900 1400 1900 2400 2900

time (sec)

Rot

(R

AD

)

0200

400600

80010001200

14001600

18002000

-0.0

47

-0.0

36

-0.0

25

-0.0

14

-0.0

030.00

80.01

90.03

0

Class

Fre

qu

ency

QS

M

-0.055

-0.045

-0.035

-0.025

-0.015

-0.005

0.005

0.015

0.025

400 900 1400 1900 2400 2900

time (sec)

Rot

(R

AD

)

0

200

400600

800

1000

1200

14001600

1800

2000

-0.047

-0.036

-0.025

-0.014

-0.003

0.008

0.019

0.030

Class

Fre

qu

ency

Fig. 14. Time histories of midspan rotational displacements and their statistic characterization (turbulent flow; V ¼ 45 m=s).


It is clear that the three formulations compute the insta-bility, but each one produces different damping for a gen-eric wind velocity. The critical wind velocities computedby the three formulations are shown in Fig. 21.

To investigate the mechanism of the change in dampingsign with the increase of wind velocity, the damping hasbeen estimated by identifying it with the exponentialcoefficient d of the function qðtÞ ¼ �q� q0 � e�d�t (�q identifiesthe static equilibrium position), which envelopes thegeneric oscillation (see Fig. 22): in damped oscillations(V < V crit), critical oscillations (V ¼ V crit) and amplifiedoscillations (V > V crit), it results d > 0; d < 0; d ¼ 0, respec-tively. The oscillations in the phase plane (rotation and ver-tical displacement) and the time projections (3D graphics)of the planes, are shown in Fig. 23. In such diagrams, theoscillations become pseudo-circular curves, which implode

in a single point (final configuration) when V < V crit, orthey stabilize themselves along a circular curve of constantamplitude when V ¼ V crit (after a transient initial periodwith different amplitude oscillations), or they explode likea divergent spiral when V > V crit.

In Fig. 24, using the QS theory, the amount of dampingd for different velocity of incident flow is shown: here, drepresents the total damping of the structural system,which is sum of the structural (assumed constant and equalto 0.5%) and the aerodynamic one (computed as the ana-lytical difference from the total damping and the structuralone). The total damping curve grows when there is anincreasing of the wind velocity; at a certain value it changesits slope and begins to decrease until the intersection ofx-axis. Such intersection represents the critical fluttervelocity.

Time history Probability density Mean values

Tra

nsv

ersa

l

0

2

4

6

8

10

12

14

400 900 1400 1900 2400 2900

time (sec)

Uy

(m)

NO_V45 ST_V45 QS_V45 QSM_V45

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

NO ST QS QSM Experim

Ver

tica

l

-3.5

-2.5

-1.5

-0.5

0.5

1.5

2.5

3.5

400 900 1400 1900 2400 2900

time (sec)

Uz

(m)


-0.4

-0.3

-0.2

-0.1

0.0


Ro

tati

on

-0.055

-0.045

-0.035

-0.025

-0.015

-0.005

0.005

0.015

0.025

400 900 1400 1900 2400 2900

time (sec)

Rot

(R

AD

)


-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0


Ro

tati

on(

DE

G)

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0


Ro

tati

on(

DE

G)

Fig. 15. Time histories of midspan displacements, statistic characterization and mean values (turbulent flow; V ¼ 45 m=s).

Vm= 21 m/s Vm= 45 m/s Vm= 57 m/s

Tra

nsv

ersa

l

NO_V21

ST_V21

QS_V21

QSM_V21

0

0.5

1

1.5

2

2.5

3

0 500 1000 1500 2000 2500 3000 3500 4000

abscissa (m)

Uy

(m)

NO_V45

ST_V45

QS_V45

QSM_V45

StaticAnalisys_V45

0

2

4

6

8

10

12

14

0 500 1000 1500 2000 2500 3000 3500 4000

abscissa (m)

Uy

(m)

NO_V57

ST_V57

QS_V57

QSM_57

0

5

10

15

20

25

30

0 500 1000 1500 2000 2500 3000 3500 4000

abscissa (m)

Uy

(m)

Ver

tica

l MA

X NO_V21

ST_V21

QS_V21

QSM_V21

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 500 1000 1500 2000 2500 3000 3500 4000abscissa (m)

Uz

(m)

NO_V45

ST_V45

QS_V45

QSM_V45

0

1

2

3

4

0 500 1000 1500 2000 2500 3000 3500 4000abscissa (m)

Uz

(m)

NO_V57

ST_V57

QS_V57

QSM_57

0

1

2

3

4

5

6

7

0 500 1000 1500 2000 2500 3000 3500 4000

abscissa (m)

Uz

(m)

Ver

tica

l min

NO_V21ST_V21

QS_V21

QSM_V21

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 500 1000 1500 2000 2500 3000 3500 4000

abscissa (m)

Uz

(m)

NO_V45

ST_V45

QS_V45

QSM_V45

-5

-4

-3

-2

-1

0

0 500 1000 1500 2000 2500 3000 3500 4000

abscissa (m)

Uz

(m)

NO_V57

ST_V57

QS_V57

QSM_57

-7

-6

-5

-4

-3

-2

-1

0

0 500 1000 1500 2000 2500 3000 3500 4000

abscissa (m)

Uz

(m)

Fig. 16. Envelopes of deck displacements (turbulent flow; V ¼ 21; 45; 57 m=s).


Transversal Vertical

Vel

oci

ties

NO_V45

ST_V45QS_V45

QSM_V45

-1.8

-0.8

0.2

1.2

0 500 1000 1500 2000 2500 3000 3500

abscissa (m)

Vy

(m/s

)


NO_V45

ST_V45

QS_V45

QSM_V45

-2.5

-1.5

-0.5

0.5

1.5

2.5

0 500 1000 1500 2000 2500 3000 3500

abscissa (m)

Vy

(m/s

)


Acc

eler

atio

ns

NO_V45ST_V45

QS_V45

QSM_V45

-0.9

-0.5

-0.1

0.3

0.7

0 500 1000 1500 2000 2500 3000 3500

abscissa (m)

ay(m

/s^2

)


NO_V45

ST_V45

QS_V45

QSM_V45

-1.5

-0.5

0.5

1.5

0 500 1000 1500 2000 2500 3000 3500

abscissa (m)

az(m

/s^2

)


Fig. 17. Envelopes of deck velocities and accelerations (turbulent flow; V ¼ 45 m=s).

0.500

0.505

0.510

0.515

0.520

600 650 700 750 800 850 900 950 1000

t (sec)

stable (positive damping)

0.500

0.505

0.510

0.515

0.520

0.525

600 650 700 750 800 850 900 950 1000

t (sec)

critical (zero damping)

0.300

0.400

0.500

0.600

0.700

600 650 700 750 800 850 900 950 1000

t (sec)

unstable (negative damping)

Fig. 18. Stable, critical and unstable oscillations.

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-10 -8 -6 -4 -2 0 2 4 6 8 10[deg]

Drag Lift Moment

Fig. 19. Polar lines used for stability problem.


5. Conclusions

In the present paper the response and the stability prob-lem of a long span suspension bridge have been studied.Four time domain approaches for aeroelastic forces formu-lations have been compared. The analyses have been con-ducted by a three dimensional complete finite elementmodel of the bridge.

Concerning the response problem one can conclude:

1. Considering nonturbulent incident wind, the differencesbetween the formulations on the structural oscillationsdamping, the QS and QSM formulations have a damp-ing greater than linear; concerning the time envelopes ofdeck displacements, the results obtained from differentformulations are very similar.

Vertical Rotational

NO

NO FLUTTER NO FLUTTER

ST

(70

m/s

)

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

580 630 680 730 780 830 880 930 980

-0.0006

-0.0004

-0.0002

0.0000

0.0002

0.0004

0.0006

580 630 680 730 780 830 880 930 980

QS

(75

m/s

)

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

580 630 680 730 780 830 880 930 980

-0.004

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

580 630 680 730 780 830 880 930 980

QS

M (

90m

/s)

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

1550 1650 1750 1850 1950

-0.0005

-0.0003

-0.0001

0.0001

0.0003

0.0005

1550 1650 1750 1850 1950

Fig. 20. Midspan unstable oscillations (V > V crit).

66m/s 70m/s 85m/s

0

10

20

30

40

50

60

70

80

90

NO ST QS QSM

V (

m/s

)

NO

FL

UT

TE

R

Fig. 21. Critical velocities (nonturbulent flow).

0.500

0.505

0.510

0.515

0.520

0.525

600 650 700 750 800 850 900 950 1000

t (sec)

teqqq ⋅−⋅+= δ0

q+

0.500

0.505

0.510

0.515

0.520

0.525

600 650 700 750 800 850 900 950 1000

t (sec)

teqqq ⋅−⋅+= δ0

Uz; Theta

q

q 0

V<Vcrit0>δ

Fig. 22. Envelope of midspan oscillation, to evaluate damping.


2. Considering turbulent incident wind, the differencesbetween the oscillations amplitude computed bydifferent formulations become significant. In general,increasing the complexity of the aeroelastic forcesrepresentation (following the succession NO, ST, QS,QSM), the maximum response decrease: this is evident

from the time histories of displacements, velocities andaccelerations, from their statistic characterization andalso from time envelopes of the deck motion. These dif-ferences increase with the increase of the wind meanvelocity.

Concerning the stability problem:

1. NO formulation cannot compute the flutter phenome-non, while the other formulations can.

Uz

The

ta

start

final

Uz

The

ta

start

final

V<Vcritδ

Uz

The

ta

start

final

Uz

The

ta

start

final

V=Vcrit0=δ

Uz

The

ta

start

final

Uz

The

ta

start

final

start

final

V>Vcritδ

>0

0<

Fig. 23. Damped, critical and amplified oscillations for midspan of the deck (phases plane representation).

Vertical Rotational

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0 10 20 30 40 50 60 70 80

Wind Velocity (m/s)

Dam

pin

g (%

)

Total Structural Aerodynamic-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0 10 20 30 40 50 60 70 80

Wind Velocity (m/s)

Dam

pin

g (

%)

Total Structural Aerodynamic

Fig. 24. Damping on incident flow velocity.


2. Increasing the complexity of the aeroelastic forces repre-sentation, the value of the critical velocity increases.

3. The variation of aeroelastic damping with the wind inci-dent velocity has been assessed using QS formulation,where the aerodynamic damping increases its value fromzero velocity to a certain value of the wind velocity;beyond this value it starts to decrease and finally itbecomes negative.

Acknowledgements

The authors thank Professors R. Calzona and K.J.Bathe for fundamental supports related to this study. Thefinancial supports of University of Rome ‘‘La Sapienza’’,COFIN2004 and Stretto di Messina S.p.A. are acknowl-edged. Nevertheless, the opinions and the results presentedhere are responsibility of the authors and cannot be


assumed to reflect the ones of University of Rome ‘‘LaSapienza’’ or of Stretto di Messina S.p.A.

References

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