A Simple Analytical Approach to the Aeroelastic Stability Problem of Long-Span Cable-Stayed Bridges

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A Simple Analytical Approach to the AeroelasticStability Problem of Long-Span Cable-Stayed BridgesGiuseppe Vairo

a

aDepartment of Civil Engineering , University of Rome Tor Vergata , Rome, Italy

Published online: 08 Jan 2010.

To cite this article:Giuseppe Vairo (2010) A Simple Analytical Approach to the Aeroelastic Stability Problem of Long-Span

Cable-Stayed Bridges, International Journal for Computational Methods in Engineering Science and Mechanics, 11:1, 1-19,

DOI: 10.1080/15502280903446846

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International Journal for Computational Methods in Engineering Science and Mechanics, 11:119, 2010

Copyright cTaylor & Francis Group, LLCISSN: 15502287 print / 15502295 online

DOI: 10.1080/15502280903446846

A Simple Analytical Approach to the Aeroelastic StabilityProblem of Long-Span Cable-Stayed Bridges

Giuseppe VairoDepartment of Civil Engineering, University of Rome Tor Vergata, Rome, Italy

This paper deals with the aeroelastic stability problem of long-span cable-stayed bridges under an approaching crosswind flow.Starting from a continuous model of the fan-shaped bridge schemewith both H- or A-shaped towers, critical states of the coupledwind-structure system are identified by means of a variational for-mulation, accounting for torsional and flexural (vertical and lat-

eral) bridge oscillations. The overall bridge dynamics is describedby introducing simple mechanical systems with equivalent stiff-ness properties and, under the assumption of a prevailing truss-like bridge behavior, analytical estimates for dominant stiffnesscontributions are proposed. Several case studies are discussed andcomparisons with experimental evidences as well as with avail-able analytical and numerical results are presented. The proposedsimplified approach proves to be consistent and effective for suc-cessfully capturing the main wind-bridge interaction mechanisms,and it could be considered as a useful engineering tool for theaeroelastic stability analysis of long-span cable-stayed bridges.

Keywords Long-span cable-stayed bridges, Aeroelastic stability,Flutter, Bridge aerodynamics

NOTATION

Ac(Aco) stay (anchor stay) cross-section area

(o) stay (anchor stay)-girder angle

B cross-section width of the deck

b halfdistance between stay curtains at the girder

level

(o) angle between stay (anchor stay) and horizon-

talx -axis

Theauthorwouldlike to thank ProfessorFrancoMaceri forvaluable

suggestions and fruitful discussions on this paper.This work was developed within the framework of Lagrange Lab-

oratory, a European research group comprising CNRS, CNR, the Uni-versities of RomeTorVergata, Calabria, Cassino,Pavia, and Salerno,Ecole Polytechnique, University of Montpellier II, ENPC, LCPC, andENTPE.

Address correspondence to Giuseppe Vairo, Department of CivilEngineering, University of Rome Tor Vergata, viale Politecnico 1.00133 Rome, Italy. E-mail: [email protected]

cA tower parameter: equal to 1 (0) for A- (H-)

shaped towers

CD, CL, CM aerodynamic drag,lift and moment coefficients

C . derivative ofC.with respect to the mean angleof wind incidence

Ct torsional stiffness of the deck

axial stay spacing along-the-chord strain variation induced in a

stay-cable by live loads

stress variation induced in a stay-cable by live

loads

D,L,M aerodynamic drag, lift and twisting moment

per unit span length

Ec(Ed) Youngs modulus of cable (deck) material

Ec (Eco) Dischingers fictitious elasticity modulus for a

stay (anchor stay)

f frequency of bridge oscillations

F the first time derivative ofF

Fi ,Fii ,Fiv the first, the second, the fourth derivative ofF

with respect to z

FI,FI I,FIV the first, the second, the fourth derivative ofF

with respect to

c specific weight of a stay

h height of the towers with respect to the deck

level

Ix (Iy ) second moment of area of the girder cross-

section around thex (y)-axis

I moment of inertia of the girder around thez-

axis per unit span length

kp(kp) flexural (torsional) stiffness at the towers top-

section

K reduced frequency of bridge oscillations undercrosswind

lc horizontal projection length of a stay

central span length of the bridge

s side span length of the bridge

m mass of the girder per unit span length

mcy(mcz) couple density about y (z)-axis due to stay-

girder interaction

1


3/20

2 G. VAIRO

Mo vertical couple acted by anchor stays upon the

towers top-section

P. , H. , A

. flutter derivatives of the deck cross-section

qcj density of stay-girder interaction forces acting

along thej-axis

qg dead loads

r1, r2 aspect ratios of the bridge:r1=

(/2h), r2=s / h

air density

s, v, w horizontal, vertical and axial displacement

components of the girder

S, U, V, W dimensionless displacement componentsof the

bridge

So axial force acted by an anchor stay upon the

towers top-section

a allowable stress in stay material

g (go ) stress in a stay (anchor stay) due to dead loads

t time variable

torsional rotation of the deck

u mean along-zdisplacement at the towers top-

section

U mean wind speed

U dimensionless mean wind speed

rotation of the towers top-section around the

verticaly -axis

circular frequency of bridge oscillations

x,y Cartesian coordinates in the plane of the girder

cross-section

z axial coordinate of the deck

dimensionless axial coordinate of the deck

1. INTRODUCTIONLong-span bridges are slender, light, and flexible large-scaleline-like structures, highly sensitive to wind effects. The main

wind-related problems affecting the behavior of such a structure

are associated with possible large deflections induced by oscil-

latory instabilities or by response to the random action of wind

gusts (buffeting). As a result of the aeroelastic interaction be-

tween wind and structure, dynamic (flutter) and static (torsional

divergence) instabilities can occur at current wind speeds.

Early studies on flutter of long-span bridges were developed

after the collapse of the First Tacoma Narrows Bridge in 1940,

and were based on the theoretical aerodynamic formulation of

airfoil flutter [13]. Bleich [4] tried to justify the Tacoma Nar-

rows collapse as a consequence of a coupled flexural/torsionalflutter by applying to the bridge the Theodorsens thin-airfoil

flutter theory. Nevertheless, the critical wind speed found by

Bleich was considerably higher than that which occurred for the

Tacoma disaster, proving that airfoil formulation is not directly

transferable to bridges. By comparing bluff-body flutter with

the thin-airfoils and involving bridge structural dynamics in

still air, Selberg [5] and Rocard [6] proposed simplified empiri-

cal formulas for estimating flutter onset velocity, widely used as

a basis of comparison among bridge stability results. These for-

mulas apply rigourously only fora flatairfoil andtheir usecan be

extended to real bridge cross-sections with a certain bluffness

amount only by employing empirical coefficients, accounting

for the actual cross-section aerodynamics and sometimes de-

pending on the bridge dynamics [7]. Nevertheless, although

these formulas are really attractive because of their simplicity,

they do not give any indication about the actual physics driving

the instability (e.g., if coupled or torsional flutter arises).

The pioneering work of Davenport [8] and Scanlan [9, 10],

among others, on bridge buffeting and flutter led to many analyt-

ical developments in bridge aerodynamics/aeroelasticity, giving

realistic descriptions of both wind-induced forces acting upon

long-span bridges and wind-structure interaction mechanisms.

From an aerodynamic point of view, Scanlan proposed a

frequency-domain model of the self-excited wind forces which

couple with the structural dynamics, by introducing frequency-

dependent empirical functions (namely, the flutter derivatives),

widely employed in specialized literature for analyzing bridge

flutter mechanisms [1122]. Generalizing the indicial con-

cepts proposed by Wagner for a flat thin airfoil [23], alterna-

tive time-domain approaches have also been proposed, based

on empirical time-dependent indicial functions of the bridges

cross-section [19, 20, 24]. Other formulations involved quasi-

steady descriptions of buffeting and aeroelastic self-excited ac-

tions, considering the static aerodynamic coefficients of the

bridges cross-section and their derivatives with respect to the

angle of wind incidence [18, 20, 25]. Nevertheless, these latter

approaches produce possible inaccurate results when wind and

structure interact with a strong unsteady character.

As far as the bridge dynamic behavior under crosswind is

concerned, the multi-mode approach, which is quite usual in

aeronautical engineering, revealed as a powerful and effectiveframework for the aeroelastic stability analysis of long-span

bridges, especially when suspension schemes are addressed.

A number of multi-mode formulations have been recently pro-

posed andthe influence of modal coupling on fluttermechanisms

has been analysed from many authors [1117]. Nevertheless,

these approaches usually need the identification of a large num-

ber of bridge modes as well as an accurate evaluation of modal

participation and interaction within the instability phenomenon.

Accordingly, they could be unsuitable and misleading in early

stages of wind-resistant bridge design.

In this context, many simplified formulations for the coupled

wind-structure problem have been proposed [21, 2629], based

on reduced dynamical systems a-priori postulated and strictlyfocused only on the simplification of the aerodynamic character-

istics of the bridges cross-section, generally not accounting for

actual and specific structural scheme of the bridge. Nevertheless,

when cable-stayed bridges are considered, specific approaches

could be conveniently applied.

As a matter a fact, stay disposition and tower shape highly

affect dynamic behavior of cable-stayed bridges. Referring

to fan-shaped stay curtains, it is possible to show that some


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AEROELASTIC STABILITY OF CABLE-STAYED BRIDGES 3

long-span cable-stayed bridge schemes with H- or A-shaped

pylons exhibit an overall truss-like dynamic behavior [30

39], mainly affected by the first flexural and torsional modes.

Accordingly, simplified dynamic models based on this truss as-

sumption can be usefully employed for analyzing the aeroe-

lastic stability of such a structure, as confirmed by many

well-established researches [3945]. Nevertheless, these studies

usually disregard the lateral deflection of the bridge and the cor-

responding non-steady wind-structure coupling, and are fully

consistent only in the case ofH-shaped towers.

In this work, generalizing the continuous model developed

by De Miranda et al. [30] and by Como et al. [46], a simplified

variational formulation is proposed for the aeroelastic stability

analysis of long-span cable-stayed bridges with fan-shaped stay

curtains, both in the case ofH- and A-shaped towers. Lateral

bridge deflection and lateral aeroelastic effects are included and,

under truss assumptions, the overall bridge dynamic behavior

is reduced to the one of simple lumped parameter systems by

introducing suitable analytical estimates of equivalent stiffness

properties. Consistence of herein employed simplified assump-

tions is proved by comparing the bridge response in still air

evaluated via the present model with analytical and numerical

solutions available in literature [33, 34, 41].

Critical conditions relevant to coupled flutter (involving cou-

pled flexural-torsional oscillations), single degree-of-freedom

flutter (namely, galloping and torsional flutter), and torsional

divergence are identified. Several applications on case studies

widely employed in literature as benchmarks and referring to

ideal or existing cable-stayed bridges are discussed, address-

ing the influence on bridge aeroelastic stability of both main

structural parameters and deck cross-section aerodynamics, as

well as of lateral effects. Finally, the comparisons with avail-

able experimental evidences and with results obtained via other

formulations prove soundness and effectiveness of the proposed

approach. As an application, clear quantitative evaluations of

the more stable behavior experienced by cable-stayed bridges

withA-shaped towers are given.

2. STABILITY OF LONG-SPAN CABLE-STAYED BRIDGES

UNDER CROSSWIND2.1. Aeroelastic Equilibrium of the Bridge: A Continuous

Model

The bridgescheme herein examined is shown in Fig. 1, where

side and central span lengths are denoted by s and , respec-

tively. A beam-like girder, axially (i.e., along the z-direction)

unconstrained, is hung to the tops of two piles, whose height

with respect to the deck level is h, by means of four plane fan-

shaped stay curtains, with a constant stay spacing . Girder

cross-section is assumed to be constant with z and the bridge

is symmetric with respect to both the vertical plane through

the z-axis and the plane orthogonal to z through the bridge

mid-span. Due to the structural symmetry of the scheme, thetwin reference system shown in Fig. 1 is adopted. Schemes

with H- and A-shaped towers (denoted in the following as HST

and AST, respectively) are addressed, assuming that pylons are

not joined with the girder. Anchor cables are connected to the

deck and at every point of the bridges ends vertical displace-

ments are prevented, so that torsional rotations are also fully

restrained. Moreover, along-x girder displacement component

is assumed to be zero at the bridges ends and at the towers

locations. The girders width B is assumed to be small in com-

parison with and, in order to take into account usually em-

ployed efficient aerodynamic cross-section designs, is different

from the distance 2b between the stay curtains at the girder

level.

FIG. 1. Long-span cable-stayed bridges with a fan scheme based onH- andA-shaped towers. Notation.


5/20

4 G. VAIRO

v(z,t)

z

yxw(t)

(z,t)

u(t)

L(t)

z

y

x

u(t)

R(t)

mcyqcy

qcz

L R

mcz

s(z,t)

qcx

FIG. 2. Displacement components and sign convention.

In accordance with the usual erection procedures (decks are

usually cantilever erected), girder and towers are assumed with

a straight reference configuration and bending free under dead

loads qg . As a consequence, stresses produced by qgcan be eval-

uated on a statically determinate truss scheme wherein hinges

are placed at stay-girder nodes [3032, 46]. As far as the ef-

fects of live and environmental (e.g., due to wind) loads are

concerned, the prevailing bridge behavior is still truss-like, thatis the arising bending and torsional stresses have a local char-

acter. Moreover, starting from the equilibrium configuration at-

tained under qg , displacements and stress variations produced by

non-dead loads can be evaluated by analyzing the incremental

response of the bridge.

Since in modern long-span cable-stayed bridges is very

small compared to and following the approach proposed in

[30, 46], an equivalent diffused stay arrangement along the deck

can be conveniently considered. Axial and shear deformability

of towers and girder as well as flexural deformability of pylons

in the plane (x, y) are neglected. Accordingly, the behavior of

the deck can be described by employing the Euler-Bernoulli

bending theory and the De Saint Venant torsion one and, undergeneral time-dependent loads, the bridge deformation is repre-

sented by the following displacement functions (see Fig. 2):

s(z, t), horizontal deflection (in the plane (x, z)) of thegirder;

v(z, t), vertical deflection (in the plane (y, z)) of thegirder;

(z,t), torsional rotation of the girder; w(t), axial (along-z) displacement of the girder; u(t), mean along-zdisplacement at the towers tops; (t), rotation of the tower top-section around the y-

axis;

where the dependency on the timethas been emphasized.

Assuming loads acting upon bridge deck to have not any

along-z component and in accordance with the stated symmetry

assumptions, the axial equilibrium of the bridge requires resul-

tant shear forces along-zat the top-sections of the towers to be

opposite. Therefore, along-zmean displacements at both tower

tops are identified by only one parameter:u.

Structural damping effects, inertial contributions of towers

and cables, wind-towers and wind-cables interactions are herein

neglected. Moreover, the bridge configuration under steady

aerodynamic loads is assumed to be practically coincident with

the one corresponding to the bridges static equilibrium under

vertical dead loads, and the wind-related bridge oscillations are

considered as a perturbation of such a configuration.

Accordingly, equations governing the linearized flexural-

torsional problem of the bridge under unsteady crosswind flow

result in:

EdIy siv(z, t) = qcx (z, t) micy(z, t)

ms(z, t) +D(z, t) (1)EdIx v

iv(z, t) = qcy(z, t) mv(z, t) + L(z, t) (2)

Ctii (z, t) = mcz(z, t) + I(z, t) M(z, t) (3)

qcz(z, t)dz kpu(t) So (t) = 0 (4)

mcy(z, t)dz kp (t) Mo (t) = 0 (5)

where i

= 4 /z

4, ii

= 2/z

2, micy

= mcy/z

and Fis the first time derivative ofF. Equations (13) denoteflexural and torsional equilibria of the girder, whereas Eqs. (4)

and (5) represent translation equilibrium along z and rotation

equilibrium around y for each tower, respectively. In Eqs. (4)

and (5) integration is performed on the stay curtains belonging

to the left (= L) or right (= R ) side of the bridge, apply-ing sign when = L and + when = R. Moreover,EdIx (EdIy ) and Ctare the bending and torsional stiffnesses

of the deck, respectively, Edbeing the Youngs modulus of the

girder;m and Iare the unit length mass and the girders polar

mass moment of inertia; qcj(z, t) denotes the density of stays-

girder interaction forces along the j-axis (j= x,y,z): mcz(z, t)andmcy(z, t) are the couple densities about z andy , induced on

the girder by the stay curtains; So (t) and Mo (t) indicate the

along-zforce and the along-y couple produced by anchor stays

at the tower tops;kpandkpare the flexural and torsional stiff-

nesses, respectively, at the tower top-sections. For H-shaped

towers made of pylons connected at the top it can be assumed

kp= kpb2, whereas for A-shaped towers kpcan be consideredapproaching infinity, so that in this caseL= R= 0.

In Eqs. (13) the unsteady wind loads per unit span length are

introduced by considering the aerodynamic lift L, drag D and


6/20


FIG. 3. Aerodynamic forces and moment: sign convention.

twisting moment M (see Fig. 3). Following a well established

approach [1820], each aerodynamic action can be described as

the superposition of: (1) a steady component; (2) a turbulent or

fluctuating component, including buffeting effects due to turbu-

lence in the approaching wind and self-induced time-dependent

components ascribable either to vortex shedding or to structure-

induced turbulence (signature effects); (3) self-excited or aeroe-

lastic force components, strongly dependent on the structural

motion. As is customary in aeroelastic analysis of long-span

bridges [1120], it will be assumed that: buffeting forces (2) arenot directly coupled with the aeroelastic ones (3), self-induced

buffeting forces are negligible, and buffeting forces due to the

turbulence in theapproaching wind areindependent on thestruc-

tural motion and they produce a negligible variation in bridge

configuration.

Accordingly, only the aeroelastic self-excited forces are re-

sponsible for the aeroelastic stability of the bridge. Assuming

that oscillations of the bridge deck under wind loads are har-

monic, being the circular frequency, the aeroelastic forces

(denoted by subscript ae) can be completely represented as

[1217]:

Dae (z, t) =1

2U2B

KP1

s

U+ KP2

B

U + K 2P3

+K 2P4s

B+ KP5

v

U+ K2P6

v

B

(6)

Lae (z, t) =1

2U2B

KH1

v

U+ KH2

B

U + K2H3

+K 2H4v

B+ KH5

s

U+ K 2H6

s

B

(7)

Mae (z, t) =1

2U2B2

KA1

v

U+ KA2

B

U + K2A3

+K 2A4v

B+ KA5

s

U+ K 2A6

s

B

(8)

where is the air density, U is the mean velocity of the on-

coming wind (generally turbulent), K= B /U is the reducedfrequency, and the eighteen real dimensionless functions ofK

denoted as Pi ,Hi ,A

i (i= 1, . . . ,6) are the flutter derivatives

for the deck cross-section. It should be observed that these lat-

ter generally depend on the z-coordinate along the deck axis

because of the possible variation of wind incidence and speed

along the bridge span as well as due to a reduction of wind

span-wise coherence induced by turbulence effects [20].

Taking into account that the bridge behavior is nonlinear

due to the intrinsic nonlinearities of the stays, the stay-girder

interaction forces introduced in Eqs. (15) can be explicitly

determined representing the mechanical behavior of a stay by

means of the Dischinger [47] and Ernsts [48] formulations (see

Appendix A).

In the framework of a truss-like behavior of long-span cable-

stayed bridges and taking into account that in these structures

the stiffness of the pylons kpis small with respect to the stiffness

of the stay curtains, it is possible to show [30, 33, 34, 39] that

the first vertical bending mode involving the displacement com-

ponent w(t) is an antisymmetric mode practically coincident

with a rigid horizontal (along-z) translation of the girder. More-

over, the second antisymmetric flexural vertical mode tends to

coincide with the first symmetrical one. Accordingly, referring

to the left side of the bridge and assuming that the aeroelastic

harmonic oscillations excited by a wind flow orthogonal to the

girders axis do not involve w (t), bridges aeroelastic stability

can be investigated by the following dimensionless form of the

dynamic equilibrium equations (15):

4y

4SI V e{[ SI + (1 cA) cAS]}I +

M

d2S

= gd2Edqg

Dae (9)

4x

4VIV + eV eU+ MV= g

EdqgLae (10)

2I I (+ cAS) + (1 cA) + SI I

= hg

Ecqgb2Mae (11)

( + X)U=

L

V d (12)

(1 cA)=1

+ X

L

( SI)d oSI|=r2(13)

where VIV = 4V/4, cA= 1 (cA= 0) for the AST (HST)scheme, and the following dimensionless quantities have been

introduced:

=z

h , d=b

h , S(, t) =s(, t)

h , (14)

V(, t) = v(, t)h

, W(t) = w(t)h

, U(t) = u(t)h

(15)

y=

4Iy g

hb2qg

1/4, x=

4

Ix g

h3qg

1/4, =

Ctg

Ecb2hqg

1/2(16)

e = EcEd

, a= 2

ch2Ec

123g, a= a 2 + /g

2(1+ /g)2 (17)


7/20

6 G. VAIRO

(a, ) = [1+ a(2 + cAd2)]1

1 + 2 + cAd2 , () = (a, ) (18)

o=2EcoAcog

Echqgsin ocos

2 osin2 o, M= m

hg

Edqg(19)

I=Ihg

Ecqgb2

, = kpgEcqg

, = L 2d + o (20)

Equations (913)generalize modelsproposedin [30, 39,40, 46],

including girderhorizontal (i.e., in (x, z) plane) deformation and

lateral non-steady aerodynamic effects, for both HST and AST

bridge schemes.

It is worth pointing out that dimensionless parameters xand

y give a measure of the ratio between the girder stiffness and

the stay-curtains one, when deck bending is considered in planes

(y, z) a n d (x, z), respectively. Analogously, measures the ratio

between the torsional deck stiffness and the stay-curtains one.

In modern long-span cable-stayed bridges, parametersx and

are small (x


8/20


where integral matrices Cae , Ks andKae are computed by the

following rule

Q(K) =

L

XTQXd

L

XTXd

1(31)

and by introducing integral flutter derivatives Pi , H

i , A

i as:

Pi (K) =L Pi SNPdL S2d

(32)

Hi (K) =L Hi VNHdL V2d

(33)

Ai (K) =L Ai NAdL2d

(34)

with NP= S, NH= NA= Vfor i= 1, 4; NP= NH= NA=for i= 2, 3; NP= V , NH= NA= Sfor i= 5, 6.

Therefore, stability limits can be found looking for wind

speeds which make singular the frequency-dependent integralimpedance matrixE(K) of the coupled wind-structure system.

It is worth observing that, in the limit of the above introduced

assumptions, the critical wind speeds should be exactly deter-

mined if E(K) is based on the exact shape of the unknown

critical mode (namely X).

Static stability limit is attained when

limK0

{det[E(K)]} = limK0

{det[Ks+ Kac (K)]} = 0 (35)

and critical coupled flutter condition is

det[2

M + iCae (K) + Ks+ Kae (K)] = 0 (36)Splitting the complex Eq. (36) in two real ones, flutter on-

set wind speed and critical circular frequency at the stability

limit can be determined by solving those equations applying a

standard iterative procedure [18].

It is useful observing that, when wind loads acting upon the

bridge deck are modelled according to a quasi-static approach,

static stability limit can be determined from Eq. (35) by putting

[25]:

limK0

K 2P3= C D, limK0

K 2P4= limK0

K2P6= 0 (37)

limK0 K2

H3= C L, limK0 K2

H4= limK0 K2

H6= 0 (38)lim

K0K 2A3= C M, lim

K0K2A4= lim

K0K2A6= 0 (39)

whereCD, CL and CMare the dimensionless aerodynamic co-

efficients [18] referred to the steady wind loads, and symbol C .denotes their first derivatives with respect to the mean angle of

wind incidence upon the bridge cross-section.

Due to Eqs. (3739), static instability can occur only when

torsional bridge response appears (static torsional divergence).

3. APPROXIMATE EVALUATION OF CRITICALCONDITIONS

As it is well known, bridge oscillations in aeroelastic critical

state can be regarded as belonging to the bridges eigen-modes

space. Therefore, due to the dominant truss-like behavior of

long-span cable-stayed bridges and in order to give estimates

of integral stiffnesses k.. defining the structural stiffness ma-

trix Ks , it is reasonable to assume that critical state is suit-

ably approximated considering only the fundamental natural

mode along each degree of freedom, i.e. S() = So(), V() =Vo(),()= o() More refined approaches could be devel-oped considering a larger number of deck modes.

In what follows it will be assumed that wind-induced bridge

oscillations produce small stress variations in stays, that is

(see Eqs. (17)) a= a and () = ().As previously highlighted, the elastic stay-girder interaction

can be assumed as negligible in comparison with the flexural

stiffness of the girder when horizontal bending is addressed.

Therefore, the integral stiffness kSS can be estimated as (see

Eqs. (31) and (78)):

kSS=4y

4

L SIVo SodL S2o d

(40)

where the first horizontal flexural mode So() can be evaluated

both for HST and AST by employing a standard approach, con-

sidering the bridge deck as a continuous beam, simply supported

at its ends and at the tower locations.

Moreover, in the framework of the truss assumptions

(x= = 0) the fundamental mode of the deck exciting verticaloscillations can be determined by using Eqs. (10) and (12) with-

out aeroelastic contributions, that is considering damping-freebridge oscillations in still air. In detail, assuming a stationary

solution as in Eqs. (21), with s= iv, Eq. (10) results in

Vo() =

1 Uo

Vo(41)

where ()= M2v/(e) and v is the natural circular fre-quency related to the fundamental vertical mode. Due to the

truss assumption is small except at the bridge midspan [34].

Accordingly, Eq. (41) can be approximated by means of a Taylor

expansion with respect to:

Vo() =Uo

Vo[1 + + 2 + 3 + o(3)] (42)

Substituting the approximated form of Eq. (41) in (12) and

solving with respect to v, different estimations of the integral

stiffnesskV Vcan be directly obtained, applicable for both HST

and AST bridge schemes. Considering linear (LA) andquadratic


9/20

8 G. VAIRO

(QA) approximations in we obtain:

k(LA)V V = 2vM=

e(+ o)0

(43)

k(QA)V V =

e0

21+ e

21

20 + 4(+ o)1 (44)

where

i=

L

2

[()]i , (i= 0, 1, 2, . . .) (45)

Moreover, when a cubic approximation (CA) is employed,

k(CA)V V can be computed solving by a standard iterative procedure

the following equation

2k3V V+ 1k2V V+ 0kV V (+ o) = 0 (46)

3.1. Bridge Scheme with H-shaped Towers

As previously pointed out, horizontal flexural oscillations

and torsional ones can be assumed to be uncoupled in the case

ofH-shaped towers, that is kS = kS= 0.Moreover, integral stiffness k can be estimated consid-

ering the bridge torsional response in still air. In detail, under

truss assumptions and considering undamped oscillations, Eqs.

(11) and (13) without aeroelastic contributions can be approxi-

mated by a Taylor expansion analogous to that used in Eq. (42).

Solving the natural circular frequency relevant to the funda-

mental torsional mode of the bridge, linear, quadratic and cubic

estimates in for kare obtained, resulting in:

k(H)

= 2I= kV V/e (47)

It is worth observing that Eqs. (10) and (11), combined with

(12) and (13), lead to

Vo() = ()

kV V/eUo

Vo, o() =

()

() kV V/eo

o(48)

that is, the first truss-like flexural and torsional modes are pro-

portional to each other.

Let the following dimensional integral stiffnesses be

introduced:

kSS= kSS Edqghg

d2, kV V= kV V Edqghg

, k= k Ecqghg

b2

(49)

and observe that, since Eq. (47), it results k(H)= kV Vb2. Ac-

cordingly, the structural dynamic behavior under crosswind can

be represented by means of the three-degrees-of-freedom cross-

section model sketched in Fig. 4a, where the stiffness of each

linearly elastic spring is defined through the dimensional quan-

tities (49), i.e. y = kV V/2 and x = kSS, attributing to the

FIG. 4. Simple lumped parameter cross-section models equivalent to the

bridge: (a) HST case, (b) AST case.

cross-section model the unit length mass and torsional inertia of

the girder (m, I).

Therefore, static stability limit condition (35) reduces to the

case of torsional divergence and the corresponding dimension-

less critical wind speed Udi v turns out to be

Udi v=Udi v

Bv=

C M

/2

(50)

where C M

results from Eqs. (34) and (39).

As far as one degree-of-freedom flutter is concerned, when

the aeroelastic damping becomes negative (namely when one of

P1 , H1 ,

A2 becomes positive) divergent oscillations can occur,leading to a damping-driven instability. Therefore, disregard-

ing aeroelastic coupling effects, critical conditions and corre-

sponding dimensionless frequencies for horizontal and vertical

galloping, as well as for torsional flutter are:

P1(Ksc ) = 0, sc=sc

v=

1 + P4(Ksc )/2(51)

H1 (Kvc ) = 0, vc=vc

v= 1

1 + H4 (Kvc )/2(52)

A2(Kc ) = 0, c=c

v=

1 + A3(Kc )/2 (53)

where K.c is the reduced frequency at the critical state and

the following ratios between natural frequencies have been

introduced:

= sv

=

kSS

kV V= d

kSS

kV V, =

v= b

m

I(54)

On the other hand, flutter instability induced by the cou-

pling of flexural and torsional divergent oscillations (namely,

stiffness-driven flutter) occurs when the complex stability


10/20


11/20

10 G. VAIRO

TABLE 1

Natural frequencies (vertical flexural fv and torsionalf) for both HST and AST schemes. Comparison among the present model

and numerical (analytical) solutions proposed in [41]. Bridge parameters: r1= 5/2,r2= 5/3,d= 0.113, = 0.434,a= 0.026,/ = 0.033,kp/qg= 50,p/qg= 0.6,e = 1,a /Ec= 7200/(2.1 106),x= 0.3,y= 4.0,= 0.05, = 0.04,= 8.

HST AST

fv (Hz) f(Hz) fv (Hz) f(Hz)

Numerical (analytical) [41] 0.272 (0.261) 0.321 (0.320) 0.272 0.429

Present model: LA 0.332 0.388 0.332 0.549

QA 0.292 0.340 0.292 0.468

CA 0.278 0.331 0.278 0.449

where

k11= x+ 2dd2

1 + d2 , k22= + 2db2

1 + d2 ,

k12

=2d

bd

1 + d2

(66)

Therefore, since the previous assumptions, Eq. (65) gives an

estimate of the natural circular frequency corresponding to the

fundamental torsional mode for the AST bridge scheme.

Applying Eq. (35) in combination with Eqs. (31), (37) and

(39), the torsional divergence onset wind speed turns out to be

Udi v=

2(2

2

d2 + 2 2d2 4) d(d 2 C

M 2C

D)

(67)

where

= 1v

x

m= d

v

kSS

M, = 1

v

I= 1

v

k

I(68)

Critical conditions corresponding to the bridge vertical gal-

loping are expressed by Eqs. (52), whereas coupled flutter in-

volving torsional and flexural (only horizontal or horizontal and

vertical) osoillations attains when a complex equation formally

equivalent to the (55) is satisfied. CoefficientsRj andGj con-

cerning these conditions are summarized in Appendix B. It is

worth while to remark that, disregarding contributions related to

flexural-torsional coupling stiffness, flutter conditions formally

reduce to those of HST scheme, in which the frequency ratio

has to be evaluated taking into account the Eq. (62).

4. VALIDATION AND APPLICATIONS

4.1. Fundamental Natural Frequencies of the Bridgein Still Air

In order to prove soundness and consistence of the proposed

integral stiffness estimates, the bridge dynamics in still air is

here addressed, considering realistic bridge schemes with H-

andA-shaped towers.

Fig. 5 depicts, versus the bridge size parameter a (see

Eqs. (17)), the frequencies fv and f of the fundamental ver-tical and torsional natural modes. Results obtained via present

formulation and relevant to different orders of approximation in

are compared with the numerical solutions proposed in [34].

These latter have been computed by nonlinear finite element

FIG. 5. Natural frequencies (vertical flexural fv on the left and torsional f on the right), versus the dimensionless bridge size parameter a for both HST

and AST schemes. Comparison among present model results (considering different approximation orders), analytical [33] and numerical [34] solutions. Bridge

parameters: r1= 5/2, r2= 5/3, d= 0.1, = 0.5, / = 0.03, kp /qg= 50, p/qg= 0.5, e = 1, a /Ec= 7200/(2.1 106), x= 0.05, y= 2.21, = 0.05, = 8.


12/20


FIG. 6. Ratio between the integral torsional stiffness of the AST bridge

scheme and the HSTs versus k(H). Bridge parameters: r 1 = 5/2, r 2= 5/3,

d= 0.1, = 0.5,a= 0.05.

analyses, accounting for the actual stay spacing and the actual

flexural and torsional stiffnesses of girder and towers. More-over, in the case of the vertical mode, present results have been

also compared with the analytical solution proposed in [33],

where nonlinear effects induced by the inertia of the towers are

included.

Further comparisons can be drawn by analyzing Table 1,

wherethe results obtained via the present approximateanalytical

approach are listed together with the analytical (not available

for the AST scheme) and numerical ones (obtained through

nonlinear finite element analyses) proposed in [41].

It can be noted that cubic estimates of kV V and k lead

to a good quantitative agreement with the reference solutions,

confirming the effectiveness of the present simplified model for

both HST and AST schemes. On the other hand, linear andquadratic approximations, although they do not give a perfect

quantitative matching with benchmark solutions, are able to

capture the main dynamic behavior of the bridge and at the same

time allow the obtainment of simpler closed-form relationships.

It is worth observing that, forgiven girderstiffness andbridge

aspect ratios, the configuration based on A-shaped towers ex-

hibits greater torsional frequencies (about 30%) than the HST-

based bridge. It clearly indicates a greater torsional stiffness of

the overall structure based on the AST scheme than the HSTs,

strictly related to the analytical estimate given in Eq. (58). In

order to furnish a quantitative indication supporting previous

considerations and referring to a bridge scheme with usual as-

pect ratios, Fig. 6 shows the ratio between the integral torsionalstiffnessk

(A)(relevant to the case AST) and k

(H)(HST) versus

k(H), for different orders of approximation in . In the range

of usual values for k(H), the AST scheme is characterized by

an overall torsional stiffness about 30150% greater than the

HSTs. For instance, employing mass and stiffness data relevant

to the bridge scheme analyzed in Fig. 5, k(H)turns out to be of

the order of 0.15 (0.181 with LA, 0.145 with QA, 0.140 with

CA) and the integral torsional stiffness k(A)is about twice.

As thiswill be remarked in the following, proposed evidences

prove that for given mass and aspect ratios, long-span cable-

stayed bridges based on the AST scheme are much more stable

from an aeroelastic point of view than those with H-shaped

towers.

4.2. Wind-induced Critical States

In order to validate the previously introduced aeroelastic sta-bility criteria and to furnish useful quantitative indications about

the influence on critical states of main structural parameters, of

bridge cross-section aerodynamics and of lateral effects, some

applications on a number of case studies will be presented,

assuming that wind forces acting upon the bridge are not depen-

dent on the axial coordinate z and neglecting any turbulence-

related contribution.

4.2.1. Sensitivity to Structural Parameters

The sensitivity of the aeroelastic critical conditions (static

and dynamic) to the structural parameters of the bridge is herein

focused. To this aim, the bridge cross-section is preliminarily

approximated, from an aerodynamic point of view, by a thin air-foil. Accordingly, assuming that a laminar crosswind acts upon

the bridge cross-section with a zero mean angle of attack and

disregarding any lateral effect, the non-zero flutter derivatives

Hi and Ai (i= 1, . . . , 4) can be expressed by means of the

Theodorsens closed-form approach [1,18]. In this case, due to

the monotonic negative behavior ofH1 andA2 with respect to

K , only a coupled flexural/torsional flutter can occur.

As generally recognized in the specialized literature

[14,18,19,21,29,39], the main structural parameters affecting

wind-induced critical states in long-span bridges are the dimen-

sionless mass ratios and , and the frequency ratio (see

Eqs. (28) and (54)) between torsional and vertical bending fre-

quencies susceptible to couple. Table 2 shows the main geomet-ric and dynamic properties, as well as the main dimensionless

parameters affecting the wind-structure aeroelastic stability, for

some existing cable-stayed bridges: Guama River Bridge, Brazil

(H-shaped towers) [50]; Seohae Bridge, Korea (H-shaped tow-

ers) [51]; Jingsha Bridge, China (H-shaped towers) [52]; Tsu-

rumi Fairway Bridge, Japan (inverted-Y-shaped towers) [29];

Yangpu Bridge, China (inverted-Y-shaped towers) [53]; Nor-

mandy Bridge, France (inverted-Y-shaped towers) [29]. It can

be observed that configurations with inverted-Y-shaped towers

can be considered as very similar to AST schemes.

Fig. 7 shows the dimensionless critical wind speed, relevant

to torsional divergence (Udi v) and coupled flutter (Uc), versus

the square of the frequency ratio and for different values of

mass ratios and . In this figure, the values of critical wind

speed obtained by the present model are put in comparison with

reference solutions available in literature. In detail, as far as

torsional divergence is concerned, the comparison is carried out

with the results obtained by the modal approach discussed in

[25], whereas critical flutter conditions are compared with those

obtained by the simplified approach proposed in [29], and by the

Selberg [5] and Rocards [6] formulas. These latter, employing


13/20

12 G. VAIRO

TABLE 2

Main geometric and dynamic properties of existing long-span cable-stayed bridges

Bridge (m) B (m) m(kg/m) I(kg m) fv (Hz) f(Hz)

Guama 320 14.2 22 513 566 838 0.331 0.649 0.01 8.0 1.96

Seohae 470 34.0 41 318 5 026 624 0.251 0.460 0.03 9.5 1.83

Jingsha 500 27.0 51 300 4 120 700 0.184 0.395 0.02 9.1 2.15Tsurumi 510 38.0 32 220 2 880 100 0.204 0.486 0.06 16.2 2.38

Yangpu 602 32.5 44 000 4 180 000 0.273 0.510 0.03 11.1 1.87

Normandy 856 23.8 13 700 633 488 0.220 0.500 0.05 12.2 2.27

the notation adopted in this paper, result in:

Uc=

0.60

(2 1)/

Selberg

1.41

(2 1)/( + 8) Rocard(69)

In order to characterize the flutter critical frequency, Fig. 8

depicts the functions 2 and2

cversus the critical reduced fre-

quency Kc. It can be observed that the critical flutter wind speed

is much more dangerous (i.e., smaller) than the torsional di-

vergence one, and for a fixed value of , critical wind speed

reduces when both and increase. Moreover, when natural

torsional and flexural frequencies tend to coincide, flutter crit-

ical wind speed strongly reduces and c tends to the natural

circular frequency of the bridge.

It can be noted that the proposed simplified approach gives

results that are in excellent agreement with the reference so-

lutions, especially for > 1. Moreover, in the case of flutter

FIG. 7. Sensitivity of the dimensionless critical wind speed for torsional divergence and coupled flutter with respect to the natural frequency ratio and mass

ratios and . Comparison among results obtained via present model, those obtained by Selberg [5] and Rocards [6] formulas, and those computed by the

simplified approach proposed in [29]. Case of a thin-airfoil-like bridge cross-section.


14/20


FIG. 8. Natural frequency ratio and dimensionless critical flutter frequency c versus the critical reduced frequencyKc. Sensitivity to the bridge mass ratios

and . Case of a thin-airfoil-like bridge cross-section.

instability, all the adopted benchmark approaches are not ableto reproduce the sudden increase in flutter critical wind speed,

appearing when the frequency ratio tends to unity and which

is strictly dependent on aerodynamic bending stiffnesses (i.e.,

on the contributions related to H4 and A4) [25]. It should also be

observed that, although the simplified model proposed in [29]

could be considered as an extension of Selberg and Rocards

formulas to generic bluff bridge cross-sections, including also

contributions related to structural damping, it is not able to con-

sider any lateral effect and is not directly based on cable-stayed

bridge dynamics.

Proposed results, obtained by using Eqs. (50) and (56), are

strictly consistent for a bridge scheme with H-shaped pylons.

Nevertheless, they can be applied also for AST, if the effectsrelated to the coupling stiffnesses kSare neglected. In this case

it is possible to show that:

(2)(A)

(2)(H)=

k(A)()

k(H)()

>1 (70)

and therefore, due to the results proposed in Figs. 6 and 7, AST

bridge scheme is proved to be less sensitive to wind-structure

effects than the HST one, resulting in a large increase of critical

wind speed.

These evidences are fully confirmed by analyzing Fig. 9,

wherein the values of the dimensional flutter critical wind speed

and of the critical frequency fcare plotted versus the main span

length , for both HST and AST configurations, and consider-ing a cable-stayed bridge scheme very similar to that studied in

FIG. 9. Dimensional flutter wind speed and critical frequency versus the main bridge span length for both HST and AST structural scheme. Case of a

thin-airfoil-like bridge cross-section. Comparison among results based on present model and the numerical solutions proposed in [41]. Bridge parameters: r1=5/2,r2= 5/3,d= 0.113, = 0.5,kp/qg= 50,p/qg= 0.6,e = 1,a /Ec= 7200/(2.1 x 106),x= 0.3,y= 4.0,= 0.05,= 0.03,= 8.


15/20

14 G. VAIRO

TABLE 3

Case studies analyzed for investigating the influence of actual

cross-section aerodynamics on critical flutter states

Case B (m) fv (Hz) f(Hz)

1. Tsurumi 38.0 0.204 0.486 0.06 16.2 2.38

2. Normandy 23.8 0.220 0.500 0.05 12.2 2.273. Seohae 34.0 0.251 0.460 0.03 9.5 1.83

4. H-shaped 12.0 0.130 0.200 0.04 3.4 1.54

5. R5 12.0 0.130 0.200 0.04 3.4 1.54

6. R5 38.0 0.204 0.486 0.06 16.2 2.38

7. R5 14.2 0.331 0.649 0.01 8.0 1.96

8. R12.5 14.2 0.331 0.649 0.01 8.0 1.96

9. R12.5 28.0 0.162 0.371 0.07 7.8 2.29

10. R20 23.8 0.220 0.500 0.05 12.2 2.27

Table 1. The proposed results (obtained including structural lat-

eral effects) highlight that both flutter onset wind speed and

critical frequency reduce when the main span length increases.

Moreover, it clearly appears that the AST scheme is much more

stable to wind effects than the HST one. For instance, for =750 m, the AST-based bridge experiences a critical wind speed

greater than the HSTs of about 75%. For that value of , the

present results arecomparedwith thenumerical solutions(based

on nonlinear finite element analyses) proposed in [41], exhibit-

ing a good agreement.

4.2.2. Influence of Bridge Cross-section Aerodynamics

As previously highlighted, thin-airfoil aerodynamics can be

deeply different from that of actual bridge cross-sections with

a bluff character. This occurrence can strongly affect wind-

structure interaction and aeroelastic stability problem. In orderto show the proposed simplified approach is able to take into

account these effects, several analyses have been carried out

considering actual bridge cross-sections under crosswind with

zero angle of incidence.

As far as dynamic flutter instability is concerned, Table 3

summarizes the different case studies analyzed, indicating the

relevant main structural parameters. In Table 4 flutter critical

conditions (critical wind speed Ucand critical frequency fc) es-

timated via present approach and available reference solutions

(or experimental data) are compared. In detail, disregarding

any lateral effects, the analyses address cases for which cross-

section geometry and flutter derivatives are available in liter-

ature: Tsurumi Fairway Bridge [18]; Normandy Bridge [54];

Seohae Bridge [55]; H-shaped cross-section [9]; rectangular

cross-sections with width/heigth ratio equal to 5, 12.5 and 20

(indicated as R5, R12.5 and R20, respectively) [21]. Proposed

results are compared with those obtained by the Selbergs for-

mula and with the well-posed approximate solutions proposed

in [29]. For the Seohae Bridge [55] and the Normandy Bridge

[54] available experimental wind tunnel test evidences are used

also as benchmarks.

Analysis of Table 4 highlights that the Selbergs formula is

no longer applicable for describing critical aeroelastic stability

conditions when actual bridge cross-sections are experienced.

On the other hand, the present simplified approach is effective

and accurate, resulting in a good agreement with the reference

solutions.

Furthermore, it is possible to observe that all the cross-

sections analyzed are prone to a coupled flexural-torsional flut-

ter, as confirmed by the monotonic negative trend ofH1 andA2 shown in the given references, except the H-shaped andrectangular R5 sections. These latter, in fact, exhibit a function

A2, which tends to reverse its sign from negative to positive atrelatively low reduced wind speed, introducing negative aero-

dynamic damping in the torsional mode and making prone thesesections to a one-degree-of-freedom torsional flutter. For these

cross-sections, resultsindicated in parentheses in Table 4 refer to

TABLE 4

Results for the case studies listed in Tab. 3. Comparisons with: the Selbergs formula, the analytical solutions proposed in [29],

some experimental evidences [54, 55]. In the case of torsional flutter: results in parentheses are relevant to the 1dof stability

criterion, results not in parentheses refer to the coupled flutter criterion

Present Benchmark Selberg

Case Uc(m/s) fc (Hz) Uc(m/s) fc (Hz) Uc(m/s) Flutter type

1 120.5 0.375 130.9 0.351 128.5 Coupled

2 81.3 0.361 >44.0 [54] 96.3 Coupled3 62.5 0.313 57.6 [55] 162.2 Coupled

4 11.5 (11.5) 0.200 (0.200) 11.5 0.200 25.4 Torsional

5 10.2 (11.4) 0.191 (0.190) 11.0 0.195 25.4 Torsional

6 58.8 (66.4) 0.370 (0.366) 58.0 0.384 128.5 Torsional

7 50.2 (51.4) 0.628 (0.626) 50.7 0.635 177.7 Torsional

8 135.4 0.559 136.5 0.560 177.7 Coupled

9 65.4 0.311 62.3 0.313 79.7 Coupled

10 88.9 0.336 87.7 0.339 96.3 Coupled


16/20


TABLE 5

Critical wind speed inducing torsional divergence. Comparison

among results computed considering actual cross-section

aerodynamics (Udi v) and those based on thinairfoil theory

(U(T h)di v )

C M

Udi v(m/s) U(T h)di v (m/s)

Guama 1.08 [50] 275.7 230.9

Jingsha 1.43 [52] 183.2 176.4

Yangpu 0.55 [53] 340.3 201.7

Normandy 1.07 [56] 129.6 106.4

the1dof stability criterionexpressed by Eq.(53), whereas results

not in parentheses are computed by the coupled flutter criterion

(56). It should be observed, in agreement with evidences pro-

posed in [16], that the coupling of bending and torsional modes

generally reduces the critical wind speed.

Anyway, trends evidenced in the previous section and related

to the influence of structural parameters on flutter conditions are

also confirmed when actual bluff cross-sections are investigated:

whenand increase critical wind speed tends to increase, and

it reduces when the frequency ratio reduces.

As far as the influence of cross-section aerodynamics on

bridge safety against torsional divergence is concerned, Table 5

reports the values of critical wind speed computed for some

of the existing cable-stayed bridges listed in Table 2. Results

obtained considering the actual cross-section aerodynamics are

compared with those relevant to the thin-airfoil theory, showing

that airfoil-based indications are always conservative and, for

actual structural stiffnesses, lead to unrealistic (very high) wind

velocities. Moreover, the comparison among values listed in

Tables 4 and 5 confirms that flutter dynamic instability can gen-

erally arise at smaller wind speeds than those inducing torsional

divergence, indicating the fundamental role of flutter as a

limiting task in wind-resistant design of long-span cable-stayed

bridges.

4.2.3. Influence of Lateral Effects and Wind Incidence

In order to further validate the proposed approach, giving

quantitative indications about the influence of lateral effects and

wind direction on the flutter stability limit, other case studies

are investigated. In detail, the existing Jingsha Bridge (China)

[52,57] and the bridge scheme addressed by Mishra et al. in

[17,58] have been analyzed.

The Jingsha Bridge over Yangtze River is a cable-stayed

bridge with a main span of 500 m and two side span of 200 m

each. The deck, in prestressed concrete, has a bluff -shaped

cross-section of 27 m wide and 2 m deep, and is hung to H-

shaped towers, of 137 m high above the deck level, by means of

two fan-shaped cable planes. Details on bridge scheme as well

as experimental-based aerostatic coefficients and flutter deriva-

tives (Hi and Ai , i= 1,2,3) at various angles of wind incidence

are given in [52]. The sign inversion (from negative to posi-

tive) of functionA2 (K) for all the angles of attack experiencedin wind tunnel tests, indicates that this bridge suffers possible

torsional flutter. This matter is confirmed by analyzing results

listed in Table 6. In detail, there are showed the natural fre-

quencies computed via proposed simplified approach, and the

critical flutter speeds obtained considering different mode com-

binations and different values of the angle of attack. It is possible

to note that the most dangerous wind attack angle is 3. Fur-thermore, if only the fundamental torsional mode participates in

flutter analysis, critical wind speed is slightly higher than that

computed considering torsional/bending (vertical and/or lateral)

coupling. Nevertheless, small differences arise, indicating that

the torsional/bending coupling is actually weak and the dynamic

instability is essentially characterized by a torsional behavior.

Anyway, the participation of the vertical bending mode induces

a lower critical wind speed, whereas flutter velocity is higher

when the lateral mode is considered. Moreover, when the not

available flutter derivatives (including the lateral ones) are ap-

proximated considering a quasi-steady approach [25], critical

wind speed slightly reduces.It should be noted that the proposed approach is conservative

and in a better agreement with the experimental-based flutter

velocity than the numerical solutions proposed in [52,57]. Nev-

ertheless, the difference among results of the present analysis

and experimental tests may be attributed to an inaccurate inter-

polation of experimental flutter derivatives, to the presence of

a certain amount of structural damping (herein neglected) and

to a possible inconsistence in assumptions leading to the bridge

TABLE 6

Jingsha Bridge: natural frequencies (Hz) of fundamental modes and, for different wind incidence, flutter critical wind speed

(m/s). Comparison among reference numerical solutions given in [52], experimental evidences [57] and results computed via

present approach (a = 0.045,p/q g= 1, cubic approximation). U(j)c : critical wind speed accounting for mode(s)j (j= s, lateralbending;j= v, vertical bending;j= , torsion). Values in parentheses refer to flutter analyses carried out approximating the not

available flutter derivatives by quasi-steady formulation

Present Ref. [52] Wind incidence U(sv)c U(v)c U

()c Ref. [52] Exp. [57]

fs 0.310 0.352 3 51.1 (49.7) 48.5 53.9 75.5 59.0fv 0.202 0.184 0

58.9 (56.4) 54.7 61.2 87.4f 0.441 0.395 +3 74.3 (72.0) 68.5 76.3 93.1


17/20

16 G. VAIRO

TABLE 7

Cable-stayed bridge analyzed in [17, 58]: natural frequencies (Hz) of fundamental modes, flutter wind speed (m/s) and critical

frequency (Hz). Comparison among reference numerical solutions given in [17] and results computed via present approach ( a =0.052,p/q g= 1, cubic approximation). Values in parentheses refer to coupled flutter analyses carried out disregarding

aerodynamic lateral effects

fs fv f Uc fc Uc fc

Present 0.145 0.176 0.310 19.2 (23.6) 0.179 (0.201) 40.1 0.301

Ref. [17] 0.157 0.167 0.280 22.3 (25.4) 0.168 (0.193) 45.4 0.296

dynamic response. It should be also pointed out that the applica-

tion of the thin-airfoil-based Selbergs formula in this case gives

a flutter speed value equal to 157 m/s, fullyin disagreement with

experimental and reference solutions.

The second case study refers to a cable-stayed bridge with a

central spanof 1020m, two sidespansof 375 m each, and with a

steel box deck 25.5 m wide and 2.3 m deep. The bridge scheme

is assumed to be characterized byA-shaped towers 224 m high

above the deck level and with a fan-shaped cable arrangement.Details of the bridge scheme and the full set of 18 experimental-

based flutter derivatives can be found in [17]. Table 7 lists the

results obtained via the present approach and put them in com-

parison with the numerical ones reported in [17]. It can be noted

that, due to a very long main span and not optimal aerodynamic

performance of the deck cross-section, the structure is very sen-

sitive to wind effects, suffering a coupled flutter for a low wind

velocity (about 20 m/s). As it can be recognized by compar-

ing critical and natural frequencies, this dynamic instability has

a dominant flexural character, both in lateral and vertical di-

rection. Moreover, the influence of lateral aerodynamic effects

clearly appears, resulting in a reduction of the computed critical

wind speed and in an increase of critical frequency, i.e. indi-cating a stabilizing character. The bridge under consideration

is also prone to torsional flutter, but for a greater wind speed

(about 45 m/s). The little percentage errors among proposed and

benchmark results can be justified as in the previous case study,

and they are surely acceptable for the first bridge design stage.

As a further comparison, considering the following data [17]

= 1.68, = 0.03, = 22.3, the flutter wind speed estimatedby the Selbergs formula for this case study is equal to 57.2 m/s,

fully in disagreement with present and reference solutions.

5. CONCLUDING REMARKS

In this paper the aeroelastic stability of long-span cable-stayed bridges has been addressed and a simplified variational

formulation for the dynamic problem of the wind-structure cou-

pled system has been proposed. Starting from a continuous

model of the fan-shaped bridge scheme with both H- and A-

shaped towers, stability limit states, with regard to both torsional

divergence and flutter, are identified by singularity conditions

of an integral wind-structure impedance matrix. This latter is

defined considering a general Scanlan-type representation of

the aeroelastic non-steady wind loads and introducing integral

stiffness properties, which allow to describe the overall dynamic

behavior of the bridge by means of simple lumped parameter

mechanical systems. Under the assumption of a prevailing truss-

like bridge behavior, integral stiffnesses have been analytically

estimated considering damping-free torsional and flexural (ver-

tical and lateral) bridge oscillations in still air. Moreover. pro-

posed closed-form relationships prove that cable-stayed bridges

withA-shaped towers exhibit torsional stiffness (deeply relatedwith the bridge sensitivity to wind effects) greater than that of

bridges based onH-shaped towers.

Several wind-structure stability analyses have been carried

out on case studies widely employed in literature as bench-

marks, referring to ideal or existing cable-stayed bridges. Clear

and useful indications from an engineering point of view have

been drawn about the influence on the bridge aeroelastic stabil-

ity of both main structural parameters and deck cross-section

aerodynamics, also considering variability of wind incidence

direction. Lateral effects, usually a-priori neglected by many

authors, have been included in proposed flutter analyses, high-

lighting that their participation in flutter mechanisms generally

cannot be arbitrarily disregarded.Furthermore, proposed results prove the ability of the present

approach to successfully capture the physics of flutter mecha-

nisms.In detail, it hasbeenemphasized that in thecase of bridges

with a streamlined deck section (similar to a flat thin airfoil),

the flutter is usually a stiffness-driven flutter, i.e. related to

the strong coupling among bending and torsional modes. On

the other hand, for bridges with deck sections characterized by

a certain bluffness amount, the flutter instability can be much

more dependent on damping-driven mechanisms, leading to

aeroelastic instabilities in which the coupling among bending

and torsional modes is weaker, and generally the torsional be-

havior is dominant. In this latter case, simplified approaches

such as those based on the Selberg and Rocards formulas arenot longer applicable, and the participating bending modes (ver-

tical and lateral) can differently affect flutter wind speed.

Finally, the agreement of the proposed results with experi-

mental evidences and with results obtained via other analytical

and numerical formulations proves the consistence and the ac-

curacy of the proposed formulation. Accordingly, it could be

considered as an effective and simple engineering tool for the

aeroelastic stability analysis of long-span cable-stayed bridges.


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APPENDIX A

The stay-girder interaction forces introduced in Eqs. (15)

are obtained by modelling the response of a stay through the

Dischinger [47] andErnsts [48] formulations. In detail, the non-

linear elastic behavior of a stay is herein modelled by means of

an equivalent fictitious elastic modulus Ec , nonlinearly depen-dent on the stress level in the cable because of the geometrical

sag effect. Accordingly, the along-the-chord stress variation

induced in a single stay by live loads p acting upon the girder

can be related to the corresponding along-the-chord strain vari-ationby the fictitious constitutive relationship= Ec [59]. Following a secant approach,Ec turns out to be:

Ec (g, ) = Ec

1 + 2

cl2c Ec

123g

2 + /g2(1+ /g)2

1(71)

where g is the stress induced in the stay by dead loads, cthe specific weight of the cable, Ec the Youngs modulus of

the cable material, lc the stay projection length in the plane

(x, z). In the limit of 0 fictitious modulus (71) reducesto the so-called tangent one and, in the framework of a small

displacement formulation, vertical bending and torsion of the

deck become uncoupled.

The equivalent distributed interaction force fbetween deck

and a stay curtain can be expressed as

f(z, t) =

Ec Ac

c (72)

where Ac is the cross-section area of the cable and the unit

vectorc identifies the stay-chord direction (see Fig. 1), defined

through the stay-deck angle and the angle between the stay

and thex -axis (= /2 for HST scheme):

c = [cos cos sin cos sin ]T (73)Assuming that stress increments in stays are proportional to

theliveloads p andthat thebridge is characterized by a dominant

truss behavior, stay and anchor stay cross-section areas (AcandAco) are fixed through the cable design stresses g (assumed

constant for all the stays) and go due to dead loads [30, 38]:

Ac=qg

2gsin ,

Aco=qg s

4gosin o

r1

r2

2 1

1 + r22+ cAd21/2

(74)

g=qg

qg + pa , go= a

1 + pqg

1

r2

r1

21

1

(75)

where a is the allowable stress, r1= /(2h), r2= s / h andindex o refers to anchor stays.

Considering, for the sake of brevity, only the left side of the

bridge, cable strain variation (z, t) results in (see Eqs. (14)

and (15)):

= 11 + 2 + cAd2

[V d(UWSIdd) cASd](76)

where sign + () applies for the front (rear) stay curtain(see Fig. 1). Accordingly, the stay-girder interactions can be

characterized by the following equalities:

qcx i + qcyj + qczk = (f(+) + f())mcyj + mczk = bi (f(+) + f()) (77)

wherei,jandkindicate the unit vectors related to the Cartesian

axes andsymbol denotes the vectorialproduct. The interactionbetween anchor stays and girder can be described following

analogous considerations.

APPENDIX B Functionsk..() introduced in Eq. (26) are:

kSS() =4y

4SI V e

2SI cAS (1 cA)

2

+

L

2SId+ oSI

=r2

I(78)

kS() = e

(1 cA)2

+ X

L

d

I(79)


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kV V() =4x

4VI V + eV e

+ X

L

V d (80)

kS() = cAS SI + (1 cA)

+ X

L 2SId

+oS

I

|

=r2 (81)

k= 2I I + (1 cA)

+ X

L

d (82)

Coefficients Rj and Gj introduced in Eq. (55) andrelevantto thefully coupled flutterof thebridge scheme

withH-shaped towers are:

R(H)o = 2 2 (83)

R(H)2 = (2 + 2 + 2 2)

2[2P4

+ 2A3 + 2 2H4 ] (84)

R(H)4 = 1 + 2 + 2 + 2 [

P4+ A3 + 2(P4+ H4 )+ 2(H4+ A3)] + o() (85)

R(H)6 = 1

2(P4+ H4+ A3) + o() (86)

G(H)2 =

2[2P1+ 2(2H1+ A2)] (87)

G(H)4 =

2

2(H1+ A2) + 2(P1+ H1 ) + P1+ A2

+ o() (88)

G(H)6

=

2

(P1+

H1+

A2)+

o() (89)

Coefficients Rj andGjoccurring in Eq. (55) when itis applied for coupled flutter involving torsional and

horizontal flexural oscillations of the bridge scheme

withA-shaped towers are:

R(As)o = 22 + 2d2 + 22 (90)

R(As)2 = (d2 + 2 + 2 + 2) +

2

d(P3+ A6)

(2 + 2)P4 (d2 + 2)A3

(91)

R(As)4 = 1 +

2(P4+ A3) + o() (92)

G

(As)

2 =

2 [d (P2+

A5) (d

2

+ 2

)A2

(2 + 2)P1] (93)

G(As)4 =

2(P1+ A2) + o() (94)

R(As)6 = G(As)6 = 0 (95)

CoefficientsRj andGjoccurring in Eq. (55) when itis applied for fully coupled flutter involving torsional

and flexural (both horizontal and vertical) oscillations

of the bridge scheme with A-shaped towers are:

R(A)o = R(As)o (96)

R(A)2 = R(As)2

2(2d2 + 22 + 22)

(1+ H4 ) (97)R

(A)4 = R(As)4 + 2 + 2 + d2

+ 2

[(2 + 2)P4 dP3+ (2 + 2 + d2 + 2)H4(d2 + 2)A3 d A6] + o() (98)

G(A)2

=G

(As)2

2

(2d2

+22

+22)H4 (99)

G(A)4 = G(As)4 +

2[(2 + 2)P1

+ (2 + 2 + d2 + 2)H1+ (d2 + 2)A2 d(P2+ A5)] + o()

(100)

R(A)6 = R(H)6 + o(), G(A)6 = G(H)6 + o() (101)

A Simple Analytical Approach to the Aeroelastic Stability Problem of Long-Span Cable-Stayed Bridges

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