Vibration Mechanisms and Controls of Long-Span Bridge -Review

7/27/2019 Vibration Mechanisms and Controls of Long-Span Bridge -Review

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248 Scientific Paper Structural Engineering International 3/2013

Peer-reviewed by international ex-perts and accepted for publicationby SEI Editorial Board

Paper received: February 8, 2012Paper accepted: January 17, 2013

Vibration Mechanisms and Controls of Long-SpanBridges: A ReviewYozo Fujino, Prof.; Dionysius Siringoringo, Res. Asst. Prof.; Bridge and Structure Lab, Civil Engineering Dept., University of

Tokyo, Japan. Contact: [email protected]

DOI: 10.2749/101686613X13439149156886

Abstract

Dynamic performance is an important consideration in long-span bridge design.Owing to its flexibility and low damping, various types of vibration from differ-ent sources of excitation could occur during the lifetime of a long-span bridge.This paper reviews important studies and developments on long-span bridgevibration mechanism and control under wind, seismic,traffic and human-motionexcitations. Types of vibration commonly observed on the long-span bridge arediscussed from the viewpoint of structure engineering. Discussion for each sub-

ject is commenced by describing the vibration mechanism followed by the surveyon observed phenomena in many long-span bridges associated with the type ofvibration. The paper also describes the engineering solutions adopted as counter-measures for each type of bridge vibration problem.

Keywords: long-span bridge vibration, vibration control, wind-induced vibration,cable vibration, seismic-induced vibration

but more extended effect such as struc-tural fatigues. In addition to that, vibra-tion can also affect user safety andcomfort and limit bridge serviceability.

In the past few decades, extensiveresearch and development has beencarried out to understand the mecha-nisms behind bridge vibration and toreduce undesirable vibration effectsthrough various countermeasures.Results of these research and devel-opments have been adopted in bridgedesign codes and put into practices byspecifying methodologies and guide-lines for countermeasures, and byintroducing new structural elementsor devices as vibration control.

In this paper, the authors reviewimportant research and develop-ments in long-span bridge vibrationwith emphasis on mechanism andcontrol. Types of vibration commonlyobserved on the long-span bridge are

discussed from structure engineeringviewpoints. Some vibration mecha-nisms are now well understood, whilesome others still require further stud-ies to achieve complete understanding.Surveys on the phenomena associatedwith the type of vibration reported inmany long-span bridges are also pre-sented as well as engineering solutionsadopted as countermeasures. Owing tospace limitation, discussions presentedin this paper focus only on bridgevibrations resulting from wind, seismic,

traffic and human-motion excitation.Interested readers are encouraged torefer to literatures cited on each sub-

ject for more detailed explanation.

Wind-Induced Vibrationof Bridges

Wind is a spatiotemporally varyingdynamic and random phenomenon,and therefore its effects on struc-ture also vary in time and space. Fora certain time period, wind speedcan be defined by the time-averaged(mean) component and the fluctuat-ing component. The total wind forceon a bridge is the summation of themean wind force (also known as thestatic component), the fluctuatingwind force due to turbulence and themotion-induced wind force. The forceapplied on bridge components causevibration in three principal direc-tions; drag force causes vibration inalong-wind direction, lift force createsvibration in cross-wind direction and

the moment force causes vibration intorsional direction. Determination ofwind forces is a difficult task as theyare sensitively influenced by the turbu-lence. In designing a bridge, the peakforce and displacement in the respec-tive directions are of primary concern.

For long-span bridges, wind-inducedvibration is the most critical amongvarious types of dynamic excitation.Type and level of vibration depend onbridge structural properties (i.e. mass,stiffness and damping), wind forcecondition and interaction between

wind and bridge. The wind-inducedvibration mechanisms in a bridge canbe classified on the basis of force exci-tation into four mechanisms: forcedvibration, self-excited vibration, com-bination of forced and self-excitedvibration and random vibration.

The bluff body of bridge componentssuch as deck, pylon and cable immersedin a wind flow can move and create amotion that in turn affects the windflow around the body. Vibration cre-ated by this interaction is known as the

Introduction

Construction of long-span bridgeshas been very active worldwide in thepast few decades. The worlds longestsuspension bridge Akashi-Kaikyo inJapan with central span of 1991 m wascompleted at the end of 20th century,and some of the cable-stayed bridges

exceeding 1000 m, such as Stonecutters,Sutong in China and Russky Bridge inRussia with central span of 1104 m werecompleted in the beginning of 21st cen-tury. A number of long-span bridgesare now under construction in China1and Korea,2 and the plans to buildsuper long-span bridges in other partsof the world are also being discussed.

As bridge spans get longer and pylonsget taller, they become more flexibleand prone to vibration. Figure 1 showshow flexible the bridge becomes as themain span increases. Flexible struc-

tures tend to vibrate under dynamicloading such as wind, earthquake,vehicle movement, and human motion.Vibration can have several levelsof consequence, from a potentiallyhazardous effect such as causing imme-diate structural failure to a more subtle


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Structural Engineering International 3/2013 Scientific Paper 249

sections use the original terminolo-gies as described in early literatures onbridge wind-induced vibration whileproviding other terminologies thathave also been used to describe thesame phenomena.

Aeroelastic Instabilities

When the structure motion and aero-dynamic force interact significantly,the amplitude of self-excited vibra-tion could grow in time with divergingcharacteristics and induce instability.This phenomenon is known as theaeroelastic instability and there arethree types of such phenomenon thatcan occur in bridge sections: torsionalflutter, coupled flutter and galloping.

Analytical formulation for studyingaeroelastic instabilities originatedfrom aerospace studies, especially insolving the aircraft flutter problem.Early formulation of motion-inducedforces on aircraft provided analyti-cal framework for experimentingwhich was further adopted for analy-sis of wind-induced vibration in bridgedecks. In Theodorsens seminal work,4which is considered by many as thefoundation of flutter analysis, the rela-tionships for unsteady aerodynamiclift force L and moment M actingon a thin airfoil are defined as linearfunctions of h (lift displacement) anda (twist angle), their time derivativesand the analytical Theodorsens com-plex circulation function C(K) = F(K)

iG(K), which is a function of reducedfrequency of oscillation K= wB/U(w:frequency of motion, B: width of thebody and U: average wind velocity).This analytical relationship can predictthe mean wind speed at which flut-ter will occur. However, implementa-tions of Theodorsens formulation tobridge decks flutter problem provedunsatisfactory. Attempts to analyze theTacoma Narrows incident as a flutterproblem by applying the Theodorsensaerodynamic formulation to the bridgefound that the estimated critical flut-ter speed was considerably higher than

what was experienced.5 Theodorsenscirculation function becomes inap-plicable as it is derived from a thinairfoil with fully attached flow whilemost bridge decks are classified asbluff bodies that experience separateflow over significant portions of theirsurface. Another study6 later sug-gested that experimentally determinedaerodynamic coefficients rather thanTheodorsens analytical coefficientscould be more useful to determineunsteady aerodynamic forces.

Fig. 2: Wind-induced vibration type with respect to wind velocity (adopted fromaeroelastic full model, wind-tunnel test of the original Tacoma Narrows Bridge,3 NV: verti-cal bending, NT: torsion)

6 24

5

Vertical1

02(

m)

Torsion

()

Vibration amplitude

Wind velocity (m/s)

1NT(vortex)

1NV(vortex)

0NV(vortex)

2NV(vortex)

3NV(vortex)

Torsionalflutter1NT

20

4 16

3 12

2 8

1

0,0

4

0,3 0,45 0,6 0,75 0,9 1 1,2 1,35 1,5

Fig. 3: Schematic figure showing wind-in-duced vibration phenomenon and its effecton bridge

Buffeting

Buffeting

Wind velocity

Vortex-inducedvibration (self-limited)

Vibrationamplitude

(verticalortorsional)

Galloping,torsionalflutter,coupled

flutter (divergent)

Fig. 1: Relationship between the lowest natural frequencies and span length obtained fromthe worlds 40 longest span suspension bridges

0

0,1

0,2

0,3

0,4

0,5

0,6

500 1000 1500 2000 2500 3000 3500

1st lateral mode

1st vertical mode

1st torsional mode

Naturalfrequency

(Hz)

Main span length (m)

Akashi-Kaikyo

BridgeProposedMessinaBridge

self-excited vibration and the resultingaerodynamic force is known as motion-dependent force. A phenomenonwherein structural motion and aero-dynamic force interact significantly iscalled aeroelastic phenomenon. Theaeroelastic phenomena commonlyobserved in long-span bridges are thegalloping, torsional flutter, coupledflutter and vortex-induced vibration

(VIV). The occurrence of aeroelasticphenomena depends on wind veloc-ity as schematically shown in Fig. 2.Moreover, wind-induced vibration andits effects on bridge can be classifiedin terms of stability, type of amplituderesponse (i.e. vibration with self-lim-ited amplitude or diverging character)and the range of wind velocity where itoccurs as illustrated by Fig. 3.

In addition to the five types of vibra-tion in this figure, this paper will alsodiscuss wind-induced vibration under

special conditions such as wake inter-ference, rainwind-induced vibrationof stay cable, dry-inclined cable gallop-ing and parametric vibration on cable-stayed bridge. One would realize thatmany terminologies have been useddifferently in describing phenomena inwind-induced vibration. The following


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dominant than the torsion results inbending flutter. Meanwhile, the tor-

sional flutter resulted when torsionalvibration is more dominant than verti-cal vibration. If the termI2 dominates,the associated stiffness-coupling insta-bility, known as the coupled flutter, isrelated to phase lag or coupling effectbetween different vibration modes.

Galloping

Galloping is defined as single degree-of-freedom large-amplitude aeroelas-tic oscillation in transverse direction. Itis also known as cross-flow galloping,translational galloping, bending flutteror cross-wind galloping. Galloping canbe experienced by prisms with certaincross section, D-sections and circularsection with some accretion such asin ice-laden cables. The amplitude ofoscillation can reach up to ten timesor more the cross-sectional dimension

of the body. In the galloping phenom-enon, the effective angle of attack ofwind velocity with respect to the bodychanges as the body experiences ini-tial motion. On some cross-sectionalshapes, this change creates differentpressure distributions that enhance theinitial motion.

Mechanism of galloping can beexplained by the quasi-steady aero-dynamic theory as explained in Refs[10,11]. Consider a bluff body underself-excited force as shown in Fig. 4undergoing wind excited vibration in

transverse (h) direction with effectiveangle of attack, a. One can define theequation of motion as:

(6)

Defining L = 1_2CLrU2B and D = 1_2CD

rU2B, total damping of the system inEq. (6) consists of structural damping

and aerodynamic damping, denoted

as ,where the terms in the bracket are eval-uated at a= 0. Considering that struc-tural damping x is positive, the system

and potential energy derivation of thesystem in Eq. (2), the time rate of thetotal energy can be obtained as9:

UTE P

Q(3)

Therefore, the increment of total

energy in the system of the bridgemotion over a period of time T0 can bewritten as:

(4)

Equation (4) implies that the totalenergy is increasing and the systembecomes unstable only when E 0.

This unstable condition is defined asflutter, and there are two necessary

conditions for the onset of flutter,namelyI1 andI2. The termI1 describesthe condition associated with diago-nal aerodynamic damping matrix andenergy due to individual single degree-of-freedom modal motion. The terminI2, on the other hand, is associatedwith the non-diagonal aerodynamicstiffness matrix and the coupled modalmotion, and is referred to as coupled

flutter.

In the aeroelastic analysis of bridgedeck, equation of motion of deckcan be described by two degree-of-

freedom vertical (h) and torsional (a)motion.

(5a)

(5b)

where xhand xa denote the structuraldamping associated with bending andtorsional motion, wh and wa are thenatural frequency of bending and tor-sional mode, respectively; while m and

I are the mass and mass moment ofinertia, respectively.

The aerodynamic forces are described

as lift force (Lh) and moment force(Ma) in vertical and torsional direction,respectively. Following description oftotal energy as previously explainedone can distinguish two conditionsfor aeroelastic instabilities, namely,the single-degree-of-freedom (SDOF)damping-induced instability in eitherbending or torsion when the term I1dominates and the stiffness-couplinginstability when the termI2 dominates.In the case of SDOF instability, a con-dition where vertical vibration is more

Extensive experimental evaluation hadbeen attempted to define unsteadyaerodynamic forces of bridge deckwithout the thin airfoil model assump-tion. Two types of experiments wereconducted: one by directly measuringthe aerodynamic force componentswhen the body is in a given specificmotion; another by calculating the force

indirectly from the induced motion ofthe body. Probably, the direct methodfor the measurement of unsteady aero-dynamic forces to bridge deck sectionswas first applied in Ref. [7]. In theexperiment, rigid bridge deck mod-els were mechanically driven into asimple harmonic motion with a rangeof specific frequency and amplitude intwo-dimensional (2D) air stream, andthe reactions at the model supportswere detected. The use of indirectmeasurement of aerodynamic forcesfor bridge decks was pioneered in Ref.[8] by detecting the induced responseof models in air flow. The indirectmeasurement generally requires less-complicated experimental set-ups thanthe direct measurement does. This tech-nique is now being widely practiced allover the world and will be described inthe following sections.

Consider a bridge girder modelledwith an N degree-of-freedom underaerodynamic force. The left-hand sideof Eq. (1) describes structural prop-erties of the bridge (i.e. mass (M),structural damping (C) and stiffness(K) symmetrical matrices), while the

right-hand-side of equation describesthe unsteady motion-dependent aero-dynamic force (i.e. FV and FD). Theterm motion-dependent implies thatthe force contains not only the instan-taneous value, but also the effectof previous motion. The forces aredefined as a function of frequency(non-dimensional frequency of oscilla-tion K= w/U), structure velocity x(t)and structural displacementx(t):

(1)

Equation (1) can be rewritten in theform:

(2)

where P = C Fv(K) and Q = K FD(K)arethe total damping and totalstiffness of the system, respectively.MatricesPand Q can be both decom-posed intoP=P1+P2 and Q = Q1+ Q2,in which the subscripts 1 and 2 denotethe symmetric and skew-symmetricmatrices, respectively. Using kinetic

Fig. 4: Schematic figure of section and windfor quasi-steady analysis of galloping

Urel

U

L

F

D

h h


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mining critical flutter wind speed wasby bimodal approach. In this approach,interaction between wind and bridgedeck is modelled as a 2D heaving (h)and twisting (a) system. The unsteadyaerodynamic forces are described as afunction of reduced frequency and thebody is assumed to undergo a simpleharmonic motion with the same fre-

quency in both heaving and twisting.This type of analysis can be tedious andcomputationally demanding, especiallyin the pre-computer days. Therefore,some simplifications were proposed toestimate the flutter speed, among whichthe most popular one for bridge deckflutter analysis is Selbergs formula22that was based on theoretical thin airfoil.

The use of bimodal approach is notalways adequate because often contri-butions of higher modes are found to besignificant. Therefore, mode-by-modeapproach is preferred for the flutter

analysis. The previous bimodal time-domain flutter analysis8 was extendedto multi-mode analysis.21 In the multi-mode analysis, the onset of flutter isdetermined by solving the complexeigenvalue problem related to char-acteristics equation of the system (see,

developments related to gallopingmechanisms in other works.13,16

Torsional Flutter

Torsional flutter is single degree-of-freedom torsional aeroelastic instabil-ity. It is also sometimes called torsional

galloping, rotational galloping or sim-

ply flutter. The mechanism of torsionalflutter was initially explained by quasi-steady theory in the linear and non-linear forms in Refs. [17, 18]. Laterstudies19,20 suggested and validatedthrough series of experiments that thetorsional flutter is in fact an unsteadyphenomenon. Therefore, unlike gallop-ing, the self-excited force for torsionalflutter is an unsteady force and mustbe described as a function of reducedfrequency (K).

Torsional flutter occurs when the totaldamping (structural plus aerodynamic)of the system in torsional motionbecomes zero. The unsteady aerody-namic forces are defined in terms of

flutter derivatives, which are the func-tion of non-dimensional frequency ofoscillation K= wB/U.8,21 The aerody-namic lift force (Lh) and moment force(Ma) are defined as:

becomes potentially unstable only ifthe aerodynamic part becomes nega-

tive, that is when .

This galloping threshold is known as

the Den Hartog criterion.12

The aerodynamic lift and drag coeffi-cients are obtained experimentally by

steady force measurement as a func-tion of angle a, and by following theabove criterion, one can obtain thecritical wind velocity for the onset ofgalloping as:

(7)

From the above derivations, it is clearthat galloping is a velocity-dependentphenomenon associated with theoccurrence of negative aerodynamicdamping. The tendency of a struc-ture to gallop can be evaluated by

checking the time-averaged lift anddrag coefficients and their signs at a= 0. Negative slope of lift force indi-cates the tendency for galloping. Thepropensity of bluff body to gallopdepends on sectional characteristics.The section with smaller height-to-width ratio in cross-flow direction haslarger tendency to gallop (soft gallop-ing), while the one with larger ratiorequires initial perturbation (hard gal-loping). Furthermore, turbulence alsocontrols the occurrence of galloping, asit changes both the lift and drag forceon the bluff body and the separation of

flow around the body. It is observed13that large turbulence intensity causeshard galloping to become soft and softgalloping to become weaker beforeeventually disappearing.

In the foregoing discussion, the criti-cal wind speed for galloping has beenpredicted according to linear theory.The non-linear quasi-steady aerody-namic theory of galloping was formu-lized in Ref. [14] using seventh powerpolynomial approximation to deter-mine the aerodynamic coefficients. Itwas found that critical velocity is the

same as that predicted by the lineartheory, only as long as the bifurcationis supercritical. In addition to criticalwind velocity, non-linear theory alsogives estimate of the amplitude andfrequency of galloping. Other stud-ies11,15 have extended the Parkinsonand Smith non-linear model14 to con-tinuous elastic structures such as can-tilevered towers and proposed theuniversal response curve that predictsgalloping of a given prismatic bodyshape. Interested readers can find

(8.a)

(8.b)

The flutter derivatives, denoted as Hi

*

(K),Ai*(K), for i = 1 to 4 are obtained

experimentally in the wind tunnel fromsectional test of bridge deck. Usingproperties of flutter derivative

A

2

*one

can estimate the critical wind velocityas the onset of torsional flutter. Thepositive values of A

2

* represent the

tendency for torsional flutter to occur.

Coupled Flutter

For most bridges, self-excited forceproduces less significant aerodynamic

coupling among modal responses,hence SDOF torsional mode gen-erally dominates the critical flutterspeed. However, there is also type ofself-excited motion known as coupled

flutter or classical flutter that occurson several degrees of freedom where,the term I2 in Eq. (4) is dominant. Inthis case, the onset of flutter dependson phase lags or degree of couplingamong modes.

In the early history of bridges flutteranalysis, the common method of deter-

for example, procedure in Refs. [23,24]). Further development includesflutter analysis for three-dimensional(3D) bridge structures, where theunsteady aerodynamic forces areapplied directly to a 3D finite elementmodel, or analyzing various 3D vibra-tion modes and assembling them usingsuperposition method. Example ofmulti-mode 3D flutter analysis of thevery long-span Akashi-Kaikyo Bridgeis presented in Ref. [25]. The analysesshow that six primary modes play sig-nificant roles in determining the fluttercondition.

As the flutter derivatives are functionsof reduced frequency, the analysis isbasically conducted in the frequencydomain. This analysis requires an itera-tive procedure to obtain the flutter crit-ical wind speed. Time-domain analysishas recently been explored by methodssuch as indicial function26 and rationalfunction approximation (RFA).27 It hasadvantages over frequency domain inthat it can include non-linear analysis


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thus are not prone to gallop. However,accretion of ice on their surface maychange the galloping characteristics.Several cases of excessive vibration onbridge cables involving ice accretionhave been reported. Among suspen-sion bridges, across-wind and along-wind vibration of the Great Belt EastBridge hanger was observed during 27

to 31 March 2001 with the maximumamplitude of 1,4 m.33 Similarly, violentvibration of stay cables observed atthe Oresund cable-stayed bridge waslikely to have been caused by snowand sleet accretion on the cable sur-face.34 A comprehensive list of thestudies related to the research on iceaccretion galloping of cables is pre-sented in Ref. [35].

Long-span bridge girders are suscep-tible to torsional flutter. It is generallyrecognized now that the dramatic col-lapse of the Tacoma Narrows Bridge

was due to torsional flutter (see Fig.2).3 The weak torsional rigidity andaerodynamically unstable girder crosssection are two main factors responsi-ble for the collapse. Long-span bridgeswith non-streamlined girder and rela-tively large width-to-depth ratio aresusceptible to torsional and coupledflutter. Since the Tacoma NarrowsBridge incident, flutter instability hadbecome the most important issue in thedesign of long-span bridges. To avoidthis instability, two different designapproaches have been proposed; one isby streamlining the girder using a box

section, this approach was promoted inthe United Kingdom in structures suchas Severn Bridge and Humber Bridge;and another using the truss-stiffenedgirder was promoted in the UnitedStates, in bridges such as New TacomaNarrows Bridge, Mackinac Bridge andVerrazano Narrows.

Pressure difference between the girderupper and lower surfaces is consideredas the main mechanism that inducesflutter instability. To reduce such pres-sure difference, part of the deck floorcan be made open using grating, or

in case of box girder, by applying thecentral slot concept.36 Effectiveness ofsuch measures depends on the locationand the opening ratio, as explained byexperiments in Refs. [37, 38] for opengrating system and central slotted sys-tem, respectively. Flutter stability ofsteel box girder can also be improvedby various aerodynamic attachmentssuch as fairings, wind nose, deflec-tors and spoilers.28,29 For truss girderbridge, the use of vertical stabilizer andcentre barrier are effective in reducing

Bridge.28 These countermeasures canreduce wind velocity of the separatedflow that originated from the leadingedge of girders lower corner, and canthus eliminate the self-excitation forceon the girder.

Tall and slender pylons are also sus-ceptible to galloping. An example of

galloping on the bridge pylon is thedamage reported on the single hexago-nal shaft of inverted Y-shape pylon ofthe Lodemann cable-stayed bridge inGermany in 1972 during strong windexcitation of about 40 m/s.31 Pylons ofcable-stayed bridges with low trans-verse beam are also prone to gallop.Pylons of the Higashi-Kobe cable-stayed bridge have this characteristic,and galloping during free-standing con-struction stage was observed in windtunnel testing.32 In Japan, aerodynamiccountermeasures are often applied toprevent galloping of bridge pylons in

the form of sectional modification suchas corner cut (Tatara Bridge) or by add-ing appendages: arch-shaped deflector(Katsushika-Harp Bridge), cover plate(Higashi-Kobe Bridge), fairing plates(Megami Bridge).28

Cables in cable-supported bridges com-monly have circular cross section and

and provide a platform for fluttercontrol analysis. However, one criti-cal issue in time-domain analysis is thedetermination of the indicial functionfrom experimentally obtained flutterderivatives, especially for non-stream-lined cross section.

Occurrences of Aeroelastic Instabilitiesand Their Countermeasures

Galloping can occur on bridge com-ponents such as girder, pylon and staycable. Non-streamlined box girderswith relatively small width-to-depthratio are particularly susceptible to gal-loping. The flow field under the bridgedeck is an important factor to controlthis phenomenon. Although cases oflarge amplitude of across-wind gallop-ing of bridge have rarely been observed,countermeasure should be provided indesign. In Japan, the galloping of bridgegirder is often avoided by means ofaerodynamic countermeasures suchas adding lower-skirts and horizontal

plate on the lower side of box girdercross section (Fig. 5). Examples of suchcountermeasures are the use of lower-

skirts at the corner of Tozaki viaductgirder (part of the Akashi-Ohnarutoline) and horizontal plate on Namihaya

Fig. 5: Examples of aerodynamic countermeasure for bridge truss and box girder againstgalloping, torsional flutter and VIV (after Refs. [2830]). (a) Aerodynamic countermea-sures for bridge deck galloping; (b) Aerodynamic countermeasures for bridge deck flutter;(c)Aerodynamic countermeasures for bridge deck VIV

Lower skirts

(a)

(b) (c)

Horizontal platesHorizontal plates

Horizontal plate

Guide vanes

Flaps

Flaps Flaps

Spoiler

Wind spoiler

Wind nose

Wind spoiler

Wind nose

Deflector

Spoiler

Open grating

Verticalstabilizer

Fairing Fairing

Fairing FairingCentral slot

Curved wind flaps

Baffles


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and on the tip of free-end pylons.52 Inbridge design, the maximum responsecaused by VIV is of particular interest.For this purpose, many formulae basedon wind-tunnel test data have beenapplied in the preliminary design. Oneexample of VIV formulation used inpractice in Japan28 is by assuming VIVas self-excited vibration, and using thesimplified lift oscillator model, theaerodynamic force is expressed as:

(9)

The formulation is used to estimate theamplitude of VIV response (y) and thecritical damping (x) that is the amountof damping that should be provided toavoid VIV. The amount of damping canbe estimated by x=rUBCL/4wnm. Theamplitude of VIV response can be cal-culated by h =rUB2C/4wnmx, where C,expressed by D/B (ratio of deck heightand width) is the self-excited lift forcecoefficient obtained from wind-tunnel

experiment.

In addition to wind-tunnel test, vortexresponse of bridge section can alsobe estimated by advanced computersimulation using computational fluiddynamics (CFD). Numerical modellingof fluidstructure interaction (FSI) inCFD is attractive for its flexibilityin usage over the wind-tunnel tests.The methods could simulate flow andbridge section response under vortexexcitation and the results look prom-ising when compared to wind-tunneltests although it may take sometime

before it can completely replace thewind-tunnel test. Examples of appli-cation of numerical approaches usingCFD to bridge engineering are thediscrete vortex method (DVM),53Reynolds average NavierStokes orRANS-based models54 and large eddysimulations (LES).55 However, CFD isyet to be proven fully effective for fullysimulating the detailed small-scaleflow features created by the deck com-ponents, especially for very stream-lined or multiple box girders.

dimension of bluff body (D) and theStrouhal number (St) (i.e. St=fvsD/U).The body oscillates under influence ofthe alternating force and the oscilla-tion remains small, unless the shed-ding frequency becomes closer to thenatural frequency of the body. In sucha case, a phenomenon known as lock-in phenomenon orshedding frequency

synchronization will occur.

The physics of VIV is very complexand finding a uniformly acceptedmodel to define its complete behav-iour remains a challenge. Nevertheless,a number of models have been pro-posed for engineering applications.The models can be classified basedon formulation of fluid force appliedon the structure in the lift direc-tion as forced system models, fluid-elastic system models and coupled

system models.16 In the forced systemmodel, the fluid force depends onlyon time and is independent of bodymotion (y), that is, F[t], such as FL(t)

= 1__2 rU2BCLsin(2PSt(u/D)t). In thefluid-elastic system models, the fluidforce does not depend only on timebut also on y, denoted as F [y(t),t].The dependence in y may include alltime derivatives, integrals and eventime delays. Example offluid-elastic

system models are the oscillator mod-els described in Refs. [40,41], empiri-cal linear and non-linear model.47 Inthe coupled system models, the fluidforce F depends on another variable(q), which depends on body motion

y, thus F[q(t), t] such as described inRef. [48]. Interested readers can findsummary of methodologies on VIV inthe following works.16,18,49,50

VIV on bridges can be classified intoat least three main categories51: one-shear-layer related vortices, two shearlayer vortices (including the Karmanvortex street and symmetrical vorti-ces) and 3D vortices. The first type isthe most commonly observed one ongirder of long-span bridges. The 3Dvortices type can be generated on theinclined cable of cable-stayed bridge

the possibility of flutter instabilities.Such countermeasures are adopted inthe Akashi-Kaikyo Bridge.28 Figure 5shows typical deck designs that consid-ered aerodynamic countermeasure forflutter.

In the design of long-span bridges, theonset of flutter should be avoided andpossibility of its occurrence should beeliminated entirely within the flut-

ter design wind velocity. For this pur-pose, wind tunnel tests on variousgirder sections and their aerodynamicattachments should be conducted andthe reference wind speed should becarefully defined by considering thedesign wind speed and wind turbu-lence effect. The design should ensurethat the critical flutter wind speed ofthe bridge under the worst conditionis still higher than the reference windspeed. Table 1 lists flutter critical windvelocity on several long-span bridgesand their countermeasures. Interestedreaders can find developments related

to flutter mechanisms and its stabiliza-tion of bridge in Refs. [30,39].

Vortex-Induced Vibration (VIV)

VIV Mechanism on Bridge Structures

VIV is one of major issues in long-span bridge vibration. Although thevibration is self-limited and doesnot create divergent type of motion,maximum amplitude, fatigue prob-lem and serviceability requirementof the bridge associated with VIV areof major concern. In finding appro-priate countermeasures to suppress

VIV, generation mechanism shouldbe firstly clarified. Generally, mecha-nisms of vortex generation can bedescribed as follows. When windpasses a bluff body, unstable shear lay-ers of opposite vorticity are produced,usually referred to as vortex shedding.The vortices create alternating peri-odic force on the body acting trans-versely to the flow direction with thefrequency fvs. The frequency is knownas shedding frequency; and its valuedepends on wind velocity (U), typical

Bridge name Main span (m) Girder type Flutter wind speed (m/s) Special flutter countermeasures

Messina42 3300 Triple box 75 Central slot

Akashi-Kaikyo43 1991 Truss 84Centre vertical stabilizer, Open Grating (centre andboth sides of girder)

Xihoumen44 1650 Twin box 78 Central slot

Great Belt East45 1624 Single box 60

Runyang44 1490 Single box 75 Centre vertical stabilizer

Tsing Ma46 1377 Single box 74 Central slot

Table 1: Examples of flutter critical wind velocity and flutter countermeasures on some of worlds longest span bridges4246


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multiple bridges and sea-crossinghighways, has a total length of 11 km,and was completed in 1997. The bridgesection includes a ten-span continuoussteel box girder with a total length of1630 m. The two longest spans of thisbridge are of 240 m. Significant vibra-tion due to vortex shedding occurredunder prevailing winds almost trans-

verse to the bridge axis during theconstruction stage. The vortex-inducedfirst-mode vibration peaked at a windvelocity around 16 to 17 m/s, withmaximum amplitude exceeding 500mm (Fig. 6).57 Extensive wind-tunneltesting using sectional models as wellas 3D models had been conductedin the design stage and the VIV wasindeed observed in the experiments.However, higher structural damp-ing and larger turbulence intensity ofthe wind at the bridge site were alsoexpected. These led to the decisionnot to install control devices fromthe beginning; instead, performanceof the bridge under wind conditionsduring the erection was closely moni-tored. It turned out that the structuraldamping and the turbulence intensityof the wind were much lower thanexpected in the design stage and theVIV indeed took place. As counter-measures, TMDs were installed insidethe box girder and aerodynamic flapswere added on the deck side beforeopening the bridge to traffic. Thesecountermeasures significantly reducedthe vibration.

VIV is also commonly observed onlong-span bridge pylons, especiallyduring construction. During the free-standing construction period, thepylons have low natural frequency andtheir bluff body would normally cause

Generally, there are two countermea-sures for VIV; one is aerodynamiccountermeasure and the other mechan-ical countermeasure using vibrationcontrol such as tuned mass. In somecases, a third option comes into playby changing the structural characteris-tics such as natural frequency or add-ing intermediate support and that is

also deemed appropriate. To suppressvibration caused by one-shear-layervortex, one needs to control generationof leading edge vortex caused by bodymotion. This can be achieved by modi-fication of leading edge of the girderusing fairing, wind nose, deflectors, andflap plates. Figure 5 shows example ofVIV countermeasures on the bridgegirder.

While adding aerodynamic append-ages can be effective in mitigating VIVof an existing bridge, the added costof installation and maintenance can

be high. Therefore, bridge girder ofan optimal shape such that it does notdisplay VIV is preferable, for exam-ple by changing the angle of inclinedpanel in a trapezoid box girder.64Controlling the pressure differenceby gratings that partially open up thedeck floor has also been used as a pos-sible stabilising method against VIV ofbridge girder. In addition, tuned-massdamper (TMD) can be installed in thebox girder to mechanically control thevibration. Table 2 lists several bridgesreported to have experienced girderVIV and their countermeasures.

The VIV problem encountered on theTrans-Tokyo Bay Bridge and its solu-tion are described here as an exam-ple. The Trans-Tokyo Bay HighwayCrossing is a combined tunnel with

Reported Occurrences of VIV onBridges and Their Countermeasures

VIV mainly occurs on bridge girderand pylon. VIV on cables of cable-supported bridge is also common; butthe amplitude is usually small and canbe suppressed completely by addinga small amount of damping. VIV on

bridge deck is more serious and usuallyaffects the girder vertical or torsionalmode. Modern long-span bridges usu-ally use box-girder deck in the trap-ezoid form. This form has superioraerodynamic instability performancethat is higher than critical wind speedfor flutter, but may suffer from VIVat low wind speed. Amplitude of VIVdepends on wind turbulence, wherelarger turbulence intensity createssmaller vibration amplitude, shape ofgirder and structural damping. Theoccurrences of VIV on the girders ofexisting bridges have been reported,

among others, in the StorebaeltSuspension Bridge,56 Trans-TokyoBay Bridge57, Second Severn CrossingBridge,58 Rio-Niteri Bridge59 andOsteroy Bridge.60 More recently, sig-nificant attention has been paid toexcessive vertical oscillations resem-bling VIV that was observed on boxgirders of the newly built VolgogradBridge in Russia.61 There has beenalso reported a case of VIV amplifiedby interference effect of two paralleldecks in Jindo cable-stayed bridge [62]Apart from cable-supported bridges,VIV can also occur on the web mem-

bers, diagonals and verticals of a long-span truss bridge. Such incident wasobserved on the vertical truss compo-nent of the Commodore Barry Bridge,the longest cantilever truss bridge inthe United States.63

Bridge name Country Max span (m) Bridge girder type Detected VIV mode VIV countermeasures

Trans-Tokyo Bay (multi-girdersteel box) [57] Japan 240 Steel box 0,34 Hz (first VB) TMD

Great Belt Suspension bridge(main span) [56] Denmark 1624 Steel box

0,13; 0,209; 0,242(third, fifth, sixth VB)

Guide Vanes (main span),TMD (approach span)

Second Severn CrossingCable-Stayed Bridge [58]

UnitedKingdom 456 Steel box 0,326 (first VB) Baffles

Rio-Niteroi Bridge (multi-girdersteel box) [59] Brazil 300 Steel twin box 0,32 (first VB) TMD

Kessock Cable-Stayed Bridge [29]United

Kingdom 240 Orthotropic deck 0,5 (first VB) TMD

Longs Creek (suspension bridge)[29] Canada I-Section 0,6 (first VB) Fairings

Volvograd Bridge (multi-girdersteel box) [61] Russia 115 steel box

0.45 Hz (first VB),0.57 Hz (second VB)and 0.68 Hz(third VB) TMD

Jindo Bridge (Twin cable-stayed)[62] Korea 340 Steel Box 0.438Hz (first VB) TMD

Table 2: Long-span bridges reported to have experienced VIV on their girder during their service (VB: Vertical Bending)


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the out-of-plane VIV. Vibration couldbe excessive and obstruct the construc-

tion process, especially for pylons madeof low-damped steel, hence vibrationcontrol is required. A well-knowninstance of VIV in tall bridge pylonsduring construction stage was in 1964,when the towers of the Forth Bridgein United Kingdom vibrated with amaximum amplitude of 1,1 m in 9 to10 m/s wind.65 VIV was also observedduring construction of Akashi-KaikyoBridge,66 the Kurushima-KaikyoBridges67 and the Storebaelt EastBridge.68 Although most of VIV on

bridge pylons are in cross-wind direc-tion, some along-wind vibrations have

also been observed, such as in theStorebaelt East Bridge68 during con-struction and on the Hakucho Bridgepylons69 after completion.

A simple method to control vibrationof bridge pylon is by connecting thetop of free-standing pylons and heavysliding blocks with cables.65 More com-prehensive control strategies employaerodynamic countermeasures suchas cutting off the corners70 or mak-ing slits, and attaching aerodynamicappendages such as deflectors or side-

plates as shown in Fig. 7. Cutting thecorner of bridge pylon reduces drag

force and cross-wind vibration atlow wind speed. This approach wasimplemented in the Higashi-KobeBridge71 and Akashi-Kaikyo Bridgepylon. Mechanical countermeasuressuch as TMD,72 tuned-liquid damper(TLD), tuned-sloshing damper (TSD)and tuned-liquid column damper(TLCD)73 are also commonly usedin recent bridges in Japan. It shouldbe noted that during construction,changes in pylon height lead to changesin natural frequency, hence tuning

Fig. 6: Observed first-mode VIV at the Trans-Tokyo Bay Bridge. Note the disappearance of a car in the red circle (afterRef. [57])


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Fig. 7: Aerodynamic countermeasures for pylon vibration control using corner cut, slit, deflector and side plate (after Ref. [70, 28])

Wind Wind

Wind

SlitDeflector

Side plate

0,50

2h/D

U/fhD

30

20

10

06/18

5/18

4/18

3/18

2/18

1/18

010

20

30

a/D

0,25Effects of section corner cut onpylon aerodynamic performance. (a:length of squared cut section,

D:length of original section,fh:heaving natural frequency,

U:wind velocity, h:heaving displacement)Critical velocity for pylon galloping increase for

a/D=2/18~3/18

Fig. 8: Vibration control measures at Akashi-Kaikyo Tower (TMD: tuned-mass damperduring and after construction is completed, HMD: hybrid-mass damper during construc-tion, after Refs. [28,66])

HMD,TMD

TMD

TMD

TMD

TMD

TMD

TMDHMD

Sensor

Damper

Weight

Spring

Servo motorsComputer

frequency should be adjusted accord-ingly. This means several mass damp-ers of different natural frequencies areneeded. To overcome this inefficiency,actively controlled mass dampers havebeen developed and this technologyhas been applied.7476 One exampleof such application is the hybrid-massdamper (HMD) used during construc-tion of the Akashi-Kaikyo Bridgepylons. HMD is a combination of pas-

sive TMD and active control actuator.The ability of this device to reducestructural responses relies mainly onthe natural motion of the TMD, whilethe forces from the control actuatorare employed to increase the effi-ciency of the HMD and to increase itsrobustness to changes in the dynamiccharacteristics of the structure. In avery long-span bridge, such as Akashi-Kaikyo Bridge, VIV is expected to

remain even after bridge completion,and therefore control countermea-sure in the form of multiple TMDs isrequired in the bridge pylons. Figure 8shows the locations of TMD and HMD.

Buffeting

Buffeting is a random bridge vibrationassociated with pressure fluctuationson the bridge due to natural wind tur-bulence. Natural wind turbulence hasvertical and horizontal fluctuationin velocity, thus analysis must takeinto account the random variationin the angle of attack. Buffeting nor-mally increases monotonically withan increase in wind mean velocity anddoes not generally lead to catastrophicfailure, but is important for evaluationof bridge serviceability. The buffet-ing forces include lift force in verticaldirection, drag force parallel to winddirection, and pitching moment.

Concepts of bridge buffeting analysishad its origin from aircraft studies. Infrequency domain, buffeting analy-sis was developed in Ref. 77, inspired

by works on thin airfoils [78, 79] byassuming and employing the quasi-steady assumption. In this approach,the spectrum of lift force is computedmode-by-mode, neglecting aerody-namic coupling and can be related tospan-wise spectrum lift force via the

joint acceptance function. Finally, thespectrum of buffeting response is cal-culated by multiplying the span-wisespectrum lift force with mechanicaladmittance function. On the basis ofthe spectrum of buffeting response,


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damping ratio (Fig. 9).98 Large cableoscillations may cause damage to theanchorage and cable fatigue. Cables ofcable-supported bridges suffer fromvarious types of vibration includingthose explained above and they arealso well documented in literature(see for example Ref 99 for a reviewand recommendation by FHWA for

stay cable100). In the following sec-tion, we shall describe special types ofvibration observed only in the cablesof long-span bridges (suspension andcable-stay), namely the wake interfer-ence vibration, rain- and wind-inducedvibration, dry-inclined cable vibra-tion and the indirectly excited cablevibration.

Wake Interference Vibration

When two circular cylinders placedin staggered arrangements in certainproximity, one windward, producing

a wake, and the other leeward withinthe wake, there is a possibility that theforces in the wake shear flow lead toa coupled instability of the submergedcylinder. Another possibility is that thevortex shedding frequency caused bythe windward cylinder coincides withfrequency of leeward cylinder creat-ing the wake-induced vortex vibra-tion of the leeward structure. The firstphenomenon can result in two typesof wake-instability depending uponthe mechanism, namely damping-controlled mechanism and stiffness-controlled mechanism. The former

mechanism relates to the gallopingcondition, in which instability occurswhen the leeward motion system hasnegative aerodynamic damping asdescribed in Den Hartog criterion. The

vulnerable construction stages shouldalso be carefully studied as shownin the case of Normandie Bridge.95Buffeting analysis requires structuralparameters, aerodynamic coefficientsobtained from wind-tunnel experi-ments and wind characteristics such asturbulence intensity, wind spectra andwind spatial correlation.

In Japanese bridge design code,28 buf-feting is taken into consideration notdirectly as the dynamic response butas the increment of static wind load.However, when fatigue damage, orcertain serviceability condition hasto be taken into account, the buf-feting maximum amplitude againstallowable amplitudes should be con-firmed. Results of structural monitor-ing system of long-span bridges havebeen used to verify buffeting analysisand to improve assumptions made incalculations.96

Cable Vibrations and Control

Cables are essential components oflong-span bridges and they are proneto vibration because of their higherflexibility, small mass and low mechan-ical damping. Main cable of suspen-sion bridge, stay cables of cable-stayedbridge and hangers are generally veryflexible compared to other bridgecomponents, and therefore prone tovibrate.97 In long-span cable-stayedbridges, stay cable can be as long as 500m, resulting in low natural frequencies

of about 0,2 to 0,3 Hz in the lowestmode. A survey on inherent dampingof stay cables of cable-stayed bridgesin Japan shows that structural damp-ing could reach as low as 0,1% critical

one can compute the peak response,which is an important parameter forbridge deck design.

The spectra analysis is popular andwidely used owing to its computa-tional efficiency and simplicity ofanalysis when dealing with unsteadyaerodynamic forces that are functions

of frequency. In addition, it providesthe necessary statistical values such asthe maximum and mean values thatare important for design. However,frequency domain approach has someshortcomings such as: disregardingof aerodynamic coupling effects,80,81limited to static coefficients that varylinearly with respect to wind angleof attack, and inadequacy in the useof strip assumption to define thespan-wise coherence of wind force.82Recent works have extended buffet-ing response estimation by spectralmethod to include the problem of cou-

pled and unsteady aerodynamic damp-ing of multi-modes system.83 Researchin time-domain buffeting analysis hasbeen very active recently, and manymethodologies have been proposed.They can be classified into four cate-gories: quasi-steady aerodynamics,8486indicial functions,8789 rational func-tion approximation27,90 and unsteadyaerodynamics.91

One critical issue in buffeting analysishas been the determination of aero-dynamic admittances. In frequencydomain, the admittance functions rep-resent the transfer functions betweenturbulence component and thesectional forces. When the quasi-steadycondition is assumed, the admittancefunctions become one. In time domain,the relationship between aerodynamicadmittances and flutter derivativesassociated with vertical motion of thedeck are described in Ref. [92], whichsuggests that the latter should be usedto estimate the former. More recently,several methodologies have been pro-posed to estimate the admittance func-tions using the flutter derivatives,93rational functions91 and indicial func-

tion.

94

Despite current developments,formulation of admittance functionsin time-domain modelling and theirfull realization in wind-tunnel testingremain a challenging task that requiressome engineering judgments and arestill subject to further research.

In long-span bridge design, buffetingmust be taken into consideration asdeformation under wind buffeting maycause structural fatigue and user dis-comfort and limit bridge serviceabil-ity to vehicles. Buffeting force during

Fig. 9: Relation between modal damping and natural frequency of stay cable (afterRef. [98])

0,3

0,1

0,0010,5

Natural frequency (Hz)

1 5

Modaldamping

(log.

decrement)

0,01

1st Mode

2nd Mode

3rd Mode

4th Mode


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not only by galloping as mentioned inRefs [109,114], but also by the influ-ence of axial flow generated by thewake and axial vortex in inclined cable(i.e. 3D vortex type). Saito et al.117 con-ducted a series of wind-tunnel test andproposed a criterion to assign certaindamping value to the stay cable sothat amplitude of cable vibration can

be reduced. Recently, two studies118,119offered a quite different interpreta-tion. In this, the flow specificity of ayawed cylinder is the main cause ofthe motion, and the presence of rain-water rivulets enhances the motion.Comparison between full-scale obser-vations and dry-inclined gallopingresults indicates noticeable similaritiesbetween the two instabilities, particu-larly the effect of the critical Reynoldsnumber regime and associated dragcrisis on both instabilities.

Despite considerable research activ-

ity over the last two decades, theunderlying physical mechanisms ofthe phenomenon are yet to be fullyunderstood. Some common under-standings based on current researchresults can be summarised as follows:(1) Vibrations often occur in flowregimes in which Karman vortex shed-ding is suppressed. (2) RWIV is relatedto, but distinct from, dry-inclined gal-loping, RWIV often occurs in the criti-cal Reynolds number range. However,the precise role of the drag crisis hasnot yet been fully established. (3) Thepresence of rivulets can act to increase

the likelihood of a galloping-type oscil-lation, however, the exact role of thesize, shape and location of the rivuletsand the effect of surface characteristicsremain to be fully understood.

Dry-Inclined Cable Vibration

Vibration with instability charac-teristics similar to RWIV has beenreported on inclined stay cables intunnel tests during dry (without rain)conditions.100,117,118120 It was shownthat if the wind is in an oblique direc-tion against the cable plane, the cable

could have similar response charac-teristics as galloping instability. Theinstability phenomenon has beenobserved in several wind-tunnel inves-tigations and it appears to share thecharacteristics, namely, occurring ondry-inclined cable at specific Reynoldsnumber range and on the few specificgeometrical cable positions. Based onwind-tunnel test, Ref. [117] describedthat unlike the RWIV, unstable con-dition can occur over a wide rangeof Scruton numbers, which means the

mechanisms. It generally occurs underreduced wind velocity above 20, andtherefore cannot be considered as VIV.As the distance between cables are toolarge and the cross section of a cablewith running water is almost identi-cal to that of a dry cable, it cannot beconsidered as wake interference andsection galloping respectively. Hence

it is considered as a new type of cablevibration caused by rain and wind.

RWIV has been observed in manycable-stayed bridges around theworld, such as Tempozan and AratsuBridge110 in Japan, Erasmus Bridge inthe Netherlands,111 Oresund Bridge inDenmark/Sweden,112 Yangpu Bridgein China and Fred Hartman Bridge inthe United States113 to name a few. Thevibration has practical consequencesthat it causes fatigue to the cableand also damage to cable anchorage.Control of this vibration becomes

one of the main concerns in long-spancable-stayed bridges.

Excitation mechanism of RWIV hasbeen the subject of many studies andunderstanding the real mechanism isvery challenging considering that thephenomenon involves not only windand cable characteristics, but alsoflowing liquid on a bluff body. Oneamong the first models that explainsthis phenomenon was proposed inRef. [114]. In this model, vibrationmechanisms are explained in twosteps. First, upper water rivulet on thecable surface are formed as a result ofa sensitive equilibrium between grav-ity, capillary and aerodynamic forces.The water rivulet effectively alters thegeometrical cross section of the cableand hence the aerodynamic forces onit. Next, depending on the locationand the size, the rivulet tends to givea negative slope of lift curve againstthe small change in the angle of attackand also significantly reduces the dragforce. These combined effects result inthe Den Hartog-type galloping insta-bility. Similar approach was taken ina model in Ref. [115], which proposes

estimation of critical wind speed andmaximum amplitude of rainwindvibration.

On the basis of results of numerouswind-tunnel experiments, Ref. 116describes three mechanisms behindthe RWIV, namely, conventionalKarman vortex, galloping instabilityand high-speed vortex excitation. Thefirst mechanism occurs at low windreduced velocity and its amplitude isrelatively small. In the second and thirdmechanisms, the instability is caused

damping-control mechanism is veloc-ity dependent and results in singledegree-of-freedom motion of the lee-ward structure and is known as wakegalloping.101,102 The later mechanism,also known as wake-flutter, is associ-ated with the coupled flutter condi-tion where oscillation occurs in twodegrees of freedom (i.e. vertical and

transverse), thus resulting in ellipti-cal orbit motion of the leeward struc-ture.103,104 The wake-flutter conditioncould occur despite the positive aero-dynamic damping condition. Note thatsome literatures refer to wake-flutteras wake galloping, and this usage israther inaccurate as galloping and flut-ter mechanisms are totally different.

Vibration of objects caused by inter-ference due to certain arrangementsand proximities are very complex innature. Several mechanisms may takeplace and this has been studied exten-

sively (see, for example, Ref. [105]).In the case of long-span cable-sup-ported bridges, wake interference hasbeen observed on the stay cables. It isobserved in the range of 2 < y/d < 2and 1 < x/d < 4, wherex andy denotethe longitudinal and lateral spacingbetween cables, respectively, and dis cable diameter.106 The oscillationstarts at the critical reduced velocity ofaround 40, which typically correspondsto the wind speed of 5 to 20 m/s, andgenerally produces an elliptical trajec-tory with the maximum amplitude lessthan 3d. The motion is known to be

sensitive to the Scruton number (Sc),and becomes hardly recognised at Sc> 50. Stay cables of Yobuko Bridge inJapan107 and the parallel suspendercables of Akashi-Kaikyo Bridge108were reported to have experienced thewake-flutter.

Rain- and-Wind-Induced Vibration

Rain- and-wind-induced vibration(RWIV) is probably the most well-known vibration problem of staycables. It was first observed by Hikamion the stay cables of the Meiko-Nishi

Bridge in Japan.109The vibration occursunder certain wind velocity range butonly during the rains. It occurs only onthe inclined cables, and the vibrationamplitude that is visible on the verti-cal plane can reach up to ten times thecable diameter. Vibration frequencyinvolves not only the first mode, butalso the higher modes. Characteristicsof RWIV are very unique such thatthe classic cable vibration mechanismssuch as VIV and wake interferencehave been disregarded as the source of


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cables in local horizontal vibration (fy)can be almost equal to the natural fre-quency of the girder global horizontalvibration (fh), a condition that leads to alinear interaction known as parametricvibration. As the cable is much lightercompared to girder and pylons, itsvibration can easily be of large ampli-tude such that the geometrical non-lin-

earity of the cable cannot be ignored.This creates the non-linear interaction.The linear internal resonance occurswhen the natural frequency ratio ofthe local cable motion to the girder-or pylon-dominant global mode is 1:1.In the forced-excitation experiment ofcable-stayed bridges, cable local vibra-tion is often excited and this is mostlydue to the linear internal resonance.Parametrically excited stay cableswere reported in Ben-Ahin Bridge,127Burlington Bridge and the SecondSevern Crossing Bridge.128

In addition to that, the frequency ofgirder global horizontal vibration (fh),can be nearly half the natural fre-quency of the girder vertical vibration(fg), which creates a condition knownas auto-parametric interaction. Thecondition where the ratio offy:fh:fg =1:1:2, is very likely in long-span, multi-stay-cable bridges, and it may occursimultaneously. A pedestrian bridge129near Tokyo is reported to have satisfiedthe above ratio between stay cablesand girders, and interaction betweencable local vibration and bridge global

vibration has been reported. Auto-parametric vibration of cables has alsobeen observed in Hitsuishijima Bridge(Fig. 11) and Tatara Bridge during theforced vibration testing [28].

Mechanisms and estimation of themaximum amplitude of vibration inter-

that the critical onset condition forthe divergent motion can be predictedquite accurately by the Den Hartog cri-terion. Although, they also noted thateffect of axial flow on the leeward sideof the inclined cable in terms of inten-sity and frequency should be taken intoaccount. Other authors have pointedout that instability clearly depends on

the critical Reynolds number range.125Further, another study126 showed thatthe main reason of inclined cable gal-loping is the difference in lift force dueto the Reynolds number, that is theterm CL/Re. The critical Reynoldsnumber is sensitive to the presenceof surface roughness, flow turbulence,motion of the cable and orientation ofthe body to the mean flow direction.

Support-Excited Vibration of StayCables

Cable vibrations are usually consid-ered local, in that the anchorage pointsat girder and pylon are fixed. On theother hand, girderpylon vibrationsare considered global, as the wholebridge span vibrates, while the cablesdo not vibrate locally but behave likemassless elastic tendons. Vibrationmechanisms in which the cables aredirectly excited by external force havebeen discussed so far. In addition tothis, motion of the girder and pylonscan also induce vibration of cablesthrough the supports or anchorages.This type of excitation is known as

support-excited excitation or indirect

excitation. When the respective naturalfrequencies of local and global vibra-tions become nearly equal to eachother, strong coupling of local andglobal vibrations may occur.

In case of long-span cable-stayed bridge(Fig. 10), natural frequency of the stay

vibration might not be suppressedsimply by adding more damping. Thework defined the criteria for instabilityknown as Saito instability line. Dry-galloping vibration could pose a veryserious problem if it takes place as pre-dicted in wind-tunnel tests. However,unlike the RWIV, incidents related tothis type of cable motion have rarely

been observed in real bridges. Perhapsa clear indication of dry galloping onstay cable was reported only recentlyin Japan,121 where a 187 m length staycable showed intense vibration withamplitude up to 1,5 m at the windspeed of 18 m/s. The large vibrationdamaged the viscous-damper at thecable-end and deck connection aswell as part of edge faring installedat the bridge girder edge. Recently, aseries of wind-tunnel tests were con-ducted by FHWA to investigate thephenomenon more closely.100 Usingthe results of the experiments, theinstability criteria was redefined suchthat large-amplitude unstable condi-tions could occur only within limitedrange of Scruton number (i.e.


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sag, bending stiffness of cable and theflexibility of damper supports on theoptimal performance of the dampers.

Vibration mitigation usually utilizessupplemental damping devices suchas high-damping rubber,137 viscousdampers,134136 and linear138 and non-linear dampers.139 In addition to that,

recently the magnetorheological (MR)damper technique has been imple-mented to cable-stayed bridges inChina, for example on the DongtingBridge140 and Shandong BinzhouBridge.141 Interested readers can findsummary of technologies for vibrationcontrol of cables on long-span bridgesin the following works (see Refs.[28,99,100,142]).

Seismic-Induced Vibrationof Bridges

Seismic-induced vibration is a forced

vibration due to inertia force pro-duced by the bridge mass and groundmotion. Structures with long naturalperiod such as long-span bridges haveless acceleration during an earthquake,but experience large displacement.Less acceleration implies less inertiaforce on the superstructure but largedisplacement means certain counter-measure to prevent pounding betweengirder and approaching span may berequired.

In the design of superstructure of along-span bridge, wind load is gener-

ally dominant; however, if large earth-quake produces unacceptable inelasticresponse, then seismic load may becomea significant factor. Design of sub-struc-tures and pylons are generally gov-erned by seismic load, while wind loadgoverns design of the girder. From aneconomic point of view, it is necessaryto identify acceptable levels of damageto a long-span bridge and to establishverification procedures for seismic per-formance against large earthquakes.

Analysis of seismic loads on long-spanbridges can be conducted in pseudo-

dynamic or static approach and infully dynamic approach depending onthe complexity of the bridge. Pseudo-dynamic analysis is a simplifiedmethod and can be divided into equiv-alent-static seismic method, multi-mode spectral method and responsespectrum method. Dynamic analysis ismore realistic as it includes 3D struc-ture, geometric and material non-lin-ear effect, multiple-support excitation,soilstructure interaction and spatialvariability of ground motion. For a

Fig. 11: Parametric vibration observed on Hitsuishijima cable-stayed bridge during forcedvibration test

185

28

HB 1 P

420 185

790

25

HB 2 P

0,7

m/s2

(a) Girder vibration : 2 Hz

(c) Cable vibration : 2 Hz

(b) Tower vibration : 2 Hz

0,7

0,7

m/s2

0,7

10

mm

10

29

HB 3 P

28

HB 4 P

action between pylon, girder and cablesin the parametric and auto-parametricexcitations are explained in some ofthe studies.130,131,132 Cable vibrationcan be very large if damping is very low.However, the possibility of having sucha vibration due to the gust response ofthe actual bridges during service maynot be high because of the positiveaerodynamic damping of the cables.

Control of Cable Vibration

Cable vibration can be controlled byaerodynamic, structural and mechani-

cal countermeasures. For RWIV, theaerodynamic countermeasures aim atbreaking up formation of the upperwater rivulet on the cable surface. Thiscan be achieved by whirling a helicalwire on the cable surface, adding dim-ples to the cable surface, or using axi-ally protuberated surface (Fig. 12). Tosuppress wake interference vibration,structural countermeasures by install-ing spacer and cable crossties betweencables are commonly applied in anattempt to increase cable stiffness.

This, however, may not be effective forvery long cables.

Currently, mechanical countermea-sure is the most widely used method.The main purpose of this approach isto increase cable damping by install-ing various types of dampers near theanchorage points or between cables.The relative motion between thecables and between cable and girdergenerate damping force.99,100,133 Thismethod, however, has a limited per-formance and adequate damping forcecannot be provided for extremely long

stay cables. The damper has customar-ily been designed by neglecting sev-eral influencing factors, one of whichis the cable flexural rigidity.133,134 It canbe anticipated that the small flexuralrigidity possessed by real cable systemsaffects performance of the damper asit is usually installed near the anchor-age. As a result, additional dampinggiven to the cable may not be as highas designed. Recent studies135,136 onthe theoretical aspects have provideddeeper understanding on the effects of


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ated with the inertial force in the caseof fixed base conditions while the quasi-static (or pseudo-static) displacementsare due to the different motions ofthe supports.143 Examples of the studyon the multiple-support excitationand spatial variability of the groundmotion of long-span bridges are givenin Ref. [144] for suspension bridge and

in Ref. [145] for cable-stayed bridge.Effect of multiple-support excitationcan be also considered in design usingresponse spectrum method by takinginto account not only the cross-modalcorrelation but also the cross-supportcorrelation.146

Type of connections between pylon andgirder affect characteristics of seismicloads on long-span bridges. Generally,shock transmission units (STU) areused as the connection between thepylons and the deck. These devices areinstalled in the longitudinal direction to

allow for expansion of the deck due totemperature changes. Under dynamicloads, these devices are extremely stiffand behave as rigid links. Various com-binations of connections such as fixed,movable, elastically constrained andunsupported or rigidly connected withpylons are possible. Fixed or rigid typeof connection will reduce the displace-ment of the deck but transfer largerinertia force of superstructure to thebase of the tower during seismic exci-tation, thus increasing the base shearand moment force on the pylons. On

the contrary, movable or floating typeconnection will result in smaller inertiaforce but requires special measure toprevent excessive deck displacement.The common approach is to compro-mise between displacements of thedeck, shear forces and moments at thebase of the pylons. Table 3 shows anexample of how connection types affectforces on the pylon and girder displace-ments of Higashi-Kobe cable-stayedbridge.71 In this case, the all moveableconnection type was selected in the finaldesign and countermeasure to limit theeffect of excessive girder displacement

was provided. Alternatively, we canuse energy dissipation devices such asactive or semi-active control systemsbetween the bridge deck and pylons.Passive energy dissipation devices canbe designed effectively for a specifiedground motion but they may be lesseffective for ground motions with dif-ferent characteristics. Effectiveness ofseismic control on long-span bridgessuch as cable-stayed bridge has been asubject of extensive research (see forexample, Ref. [147]).

Fig. 12: Aerodynamic countermeasures for cable vibration control at (a) Yobuko Bridge(Cross-tie), (b) Akashi-Kaikyo Bridge (helical stripes), (c) Higashi-Kobe Bridge (axiallyprotuberated surface) and (d) Tatara Bridge (indented surface)

Cablecross-tiesystem

(a) (b)

(d)(c)

Indentedcable

surface

Axiallyprotuberatedcable surface

Helical stripes

Connection between pier/pylonand girder

Frequency(Hz)

Force at the pylon base103 kN, kN m

DisplacementGirder (mm)

Two connection fixed (MFFM) 0,7 M= 608; N= 90; S = 24 200

All connections fixed (FFFF) 0,79 M= 609; N= 85; S = 24 180

Elastic spring connections(MSSM) 0,33 M= 308; N= 88; S = 10 370

Moveable connections(MMMM) 0,12 M= 155; N= 90; S = 2 560

One connection fixed(MFMM) 0,45 M= 602; N= 97; S = 23 220

Load combination: dead load + live load during earthquake + earthquake load (M: moveable,F: fixed).

Table 3: Examples of how type of pier/pylon-to-girder connections affect the force anddisplacement (Higashi-Kobe Bridge, after Ref [71])

long-span bridge, multiple-supportexcitation and spatial variability of theground motion become significant fac-tors in the seismic response.

Spatial variability of ground motions ispossible considering that input forcesfrom multiple supports separated byfew hundred meters can be differentdepending on the local site condition.The contributing factors to spatial vari-ability of ground motions are dynamic

properties of surrounding soils, wavepassage effects such as propagationspeed, distance between supportsand effects of incoherence in the seis-mic wave. The common approach forincluding multiple-support excitationand spatial variability of the groundmotion in the analysis of long-spanbridges is by decomposing the totalstructural response into dynamic com-ponent and pseudo-static component.The dynamic displacements are associ-


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of trainbridge interaction, in addi-tion to the problems mentioned above.They are train runability, safety andpassenger comfort.160 Train runabilityconcerns with the vertical slopes andhorizontal planes of the railroadonwhich the train can operate safely. Forthe long-span bridges, train runabilitymay be affected by bridge deformation

not only because of gravitational loadbut strong wind and seismic actionas well, and these conditions must befully investigated during design. Safetyissues concerning possible derailmentduring extreme events such as strongwind and earthquake and their effectson bridge and train have to be lookedinto carefully.

Human-Induced Vibrationof Pedestrian Bridges

Human-induced vibration is a prob-lem commonly observed on pedes-trian bridge, although, recently someof vehicle bridges too are reportedto have experienced human-inducedvibration during events such as a mar-athon or a rally. The topic is includedin this paper as it is considered as animportant vibration problem thatshould be addressed while designing along-span pedestrian bridge.

There have been many reports ofhuman-induced vibration in pedestrianbridges in the past, but the problem hasattracted considerable attention from

bridge engineers and researchers afterthe infamous incident of MillenniumBridge in London in year 2000.Several studies161163 have presenteda comprehensive review on reportedcases of human-induced vibration onpedestrian bridges and recent researchand developments on the remedies.The problem has attracted particularinterest from research communitiesin European countries as pedestrianbridges are very popular in this region.In general, human-induced vibrationis a serviceability issue rather than a

safety concern, and it becomes an issuemainly because human beings arevery sensitive to vibration. However,if vibration occurs very frequently,fatigue may become a concern regard-ing service lifetime of the bridge.

Human-induced vibration of pedes-trian bridge may occur in verticaland lateral directions. There are twoissues in humanbridge interactionconcerning pedestrian bridge, onebeing the interaction between humanmotion and the bridge, and the other is

bridges load-bearing capacity. Inaddition to gravitational vehicle load,vehicle movement can introduce loadamplification commonly known asdynamic amplification factor (DAF).The DAF depends on three main fac-tors, namely, bridge structure (i.e. typeof bridge, natural frequency, supportand joint condition), vehicle character-

istics (i.e. weight, speed and dynamiccharacteristics) and pavement condi-tions (i.e surface roughness and trans-verse path). DAF has been studiedcomprehensively and described invarious bridge design codes, althoughconsensus of its value and methodsremain a topic of discussion. A com-prehensive review of research anddesign code of the DAF is providedin Ref. [155]. Traffic-induced vibrationmay create local stress fluctuation onthe bridge. Large stress fluctuation dueto traffic may induce structural fatigue

at certain structural components, espe-cially steel bridges that may reducethe expected service life of the bridge.Fatigue of steel bridge componentshas been extensively investigated; Ref.[156], for example, presents a compre-hensive list of studies on this subject.

Vehiclebridge interaction is animportant factor in analysis of traf-fic-induced bridge vibration. In theearly studies on this topic, the inter-action problem was simplified usingmoving-force and moving-mass modelon the simple bridge model to obtainfundamental understanding.157,158Nowadays, with the advancement ofdigital computer, more realistic vehicleand bridge models can be realised andthe effect of interaction can be betterunderstood.159

The effect of vehiclebridge interac-tion is especially important for railwaybridges, where large kinetic energycarried by trains at high speeds mayinteract significantly with a bridge andeven resonate with it under certainconditions. Vertical vibration caused bytrain speed will induce excessive vibra-tion in the columns of the viaducts sup-porting the rails. The lateral vibrationis of primary concern with regard tothe safety of the passengers to preventderailment. Resonance between a sup-porting bridge and high-speed trainmay occur under certain train speedand bogies spacing. And these condi-tions are generally found in the shortto medium railway bridges having spanlength between 10 and 40 m.

For long-span bridges, there are at leastthree important issues in the analysis

Generally, designs of long-span cable-stayed bridges in Japan employ moreflexible type of connections that pro-duce less seismic force while con-trolling displacement with specialdevices such as148: short tension links(Yokohama-Bay Bridge149), elasticconstraint (Nishi Meiko Bridge), largesprings (Hitsushi/Iwakuro Bridge)

and long-vertical suspended cable(Higashi-Kobe Bridge). In the Rion-Antirion Bridge,150 a different strat-egy was adopted. In this bridge, windand static loads are transferred to thepier in transverse direction using a stiffmetallic fuse device. Parallel to thissystem, a number of viscous dampersare placed and designed to dissipatelarge amounts of energy after the fusesyield in the event of large earthquake.

Although many long-span bridges arebuilt in seismically active regions, onlyfew are known to have been dam-

aged by earthquakes. Among themare: the Higashi-Kobe cable-stayedbridge that suffered from end-linkfailure during the 1995 Hanshin KobeEarthquake, Japan; eastern span of theSan FranciscoOakland Bay Bridge,a cantilever truss bridge whose uppertruss collapsed during the 1989 LomaPrieta earthquake; Ji Ji Da cable-stayed bridge in Taiwan that sufferedfrom damage on the main pylon andthe main girder during the 1999 Chi-Chi earthquake; and Shipshaw cable-stayed bridge in Canada during the

1993 Saguenay earthquake.Recently, many long-span cable-sup-ported bridges in seismically activeregions have been instrumentedwith vibration sensors to investigatestructural responses during largeearthquakes. Seismic responses ofthese bridges were analyzed duringlarge earthquakes such as during the1987 Whittier Narrows and the 1994Northridge earthquake for VincentThomas Suspension Bridge151, the1995 Hyogo-Nambu earthquake forHigashi-Kobe Bridge152 and the recent2011 Great East Japan earthquake forYokohama-Bay Bridge.153 They haveenhanced our understanding in the non-linearity aspects of long-span bridgeresponse,151,152 soilstructure interac-tion152 and performance of structuralconnections during earthquakes.153,154

Traffic-Induced Vibrationof Bridges

The main concern related to traffic-induced vibration is its effect on the


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ous models have been proposed todescribe the pedestrian loading andsynchronization mechanism. Somemodels assumed that synchronizationis the cause of lateral vibration, suchas in Refs. [168, 169]. The models useparameters whose values are fittedwith experimental data. The case ofMillennium Bridge lateral vibration169was used to model the human walkingforce on the bridge as forced-excitation

mechanism by separating the naturalmovement of the centre mass on a sta-tionary pavement x(t) and movementof the pavement y(t), and examinedthe influence of time lag () betweenpeople reaction and the bridge move-ment, and some constants aandb. Thehuman walking load force is modelledas:

(11)

Furthermore, the study investigatesthe effect of people-to-bridge massratio on the stability of bridge vibrationand evaluates the amount of requireddamping to suppress the vibration.

The pedestrian force and synchro-nization process are described usinga model adopted from mathemati-cal biology in Refs. [170, 171]. In thismodel, each pedestrian (i) exerts alter-nating sideways force F(t) defined as:

, where G is the maxi-

mum force and the phase increase by2 during a full sideways walking cycle.In turn, pedestrian walking pattern is

on the bridge, caused by Nnumber ofpeople with specific pedestrian lateralforce (p), walking frequency (wp) andrandom phase (fi):

(10)

The bridge lateral vibration wasthought to be caused mostly by directlinear resonance (Fig. 13) between

lateral force of human walking andthe girder lateral mode. The simpleforced-excitation model of humanwalking can explain qualitatively theoccurrence of lateral motion but can-not explain the large lateral vibrationobserved on the bridge. The model wasunable to explain explicitly the growthprocess of lateral vibration, but syn-chronization process of human walk-ing is assumed to increase the humanforce and to have caused the bridgeresonance.

In a subsequent research, a two-legrigid rocking model was proposed inRef. [167] to describe pedestrian walk-ing. Lateral force is calculated by thismodel and interaction between pedes-trian walking and girder lateral vibra-tion is considered under assumptionthat phase lag exists between humanmotion and walking force. The resultsof the calculation agree well withexperiments of human walking on alateral shaking table.

More recently, especially after theMillennium Bridge incident, vari-

synchronization of movement amongpedestrians as input force to the bridgevibration. Regarding the first issue, itis well known that human walkingproduces vertical periodic force thatexcites vertical motion of the girders.Frequency of this force is about 2 Hz(two steps per second), and designcode in Japan for instance specifies

that the natural frequency of the verti-cal modes of pedestrian bridge shouldbe outside of the 1,7 to 2,5 Hz rangeto avoid the resonance vibration of thegirder.164 In addition to vertical vibra-tion, human walking also produceshorizontal periodic force, becausehuman beings use two legs in walkingand its frequency is around 1 Hz. Thelateral force of human walking canpotentially excite lateral motion of thegirder if the natural frequency of thelateral modes is close to 1 Hz.

Synchronization of PedestrianMovement

Synchronization of movement amongpedestrians can occur in vertical andlateral directions as they are passingthe bridge. While synchronization invertical direction is much less likely,synchronization in lateral direction canoccur and becomes more pronouncedwith larger vibration.

One of the first studies that investigatesuch vibration is the lateral vibrationobserved on T-Park Bridge in Japanduring congested passage of pedestri-

ans.129 Analysis of recorded bridge andhuman vibration motion through videoof pedestrians head motions revealedthat pedestrians who noticed the lat-eral sway of the bridge attempted tore-establish his/her balance by mov-ing his/her body in the opposite direc-tion. By doing so, he or she exertedload on the pavement and enhancedstructural vibration. The frequency ofhuman walking tends to synchronizeand becomes lock-in to the frequencyof bridge lateral vibration. This excitesthe bridge in a resonant manner, result-

ing in a large lateral girder vibrationwith amplitude of several centimeters.This phenomenon is mentioned inRef. 165 assubconscious adjustment ofindividual pedestrian to any vibrationof the pavement that leads to feedbackand synchronization. Similar phenom-enon was observed in Solferino Bridgein Paris and Millennium Bridge inLondon during its opening day in year2000.166

In the case of T-Park Bridge,129 humanwalking is modelled as periodic force

Fig. 13: T-Park Pedestrian Bridge near Tokyo experienced lateral synchronized human-induced vibration and the two excitation mechanisms (after Ref. [129])

Mechanism:

Viewed from above

1cmc.g.

1,0 s

1 Hz 1 Hz

1 Hz


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ing a suitable cure for the problem.Through decades of research, we nowunderstand the mechanisms behindmost of vibration problems com-monly observed in long-span bridges.Vibration problems whose mecha-nisms are yet to be fully understoodare still present which require furtherstudies. Damping, for example, is a very

important component in the vibrationof long-span bridges, yet its quantifi-cation remains challenging. Structuraldamping is expected to be smaller forlonger span bridges; therefore, mecha-nisms to increase damping will becomemore important for future design.

Countermeasures against vibrationproblems have been developed on thebasis of the present understanding ofvibration mechanisms. Performanceimprovement for these engineeringcountermeasures is still a subject forfurther study, especially for bridge

with longer spans and taller pylons.One particular aspect of the counter-measures is the use of control tech-nology for long-span bridge vibrationproblems. While passive control strate-gies have been widely applied, activecontrol strategies have yet to reach amature state to be directly applied tosolve long-span bridge vibration issues,which provides both challenges andopportunities for bridge researchersand engineers in the future.

Recent advancements in computertechnology have provided us with

strong simulation tools to investigatevibration problems. CFD technique,for example, has been developed andused extensively as a supplement inrecent years. With rapid development,CFD is expected to become a strongtool, although it may take some timeto become a complete replacementfor wind-tunnel testing because of thehigh complexity of the geometry ofthe bridge section and the 3D natureof the bridge. For seismic-inducedvibration analysis, the use of computersimulation improves our understand-ing of the behaviour of bridge com-ponents under specific earthquakes,which is essential in seismic perfor-mance-based design.

New developments and technologiesin sensing and monitoring have beenapplied in modern long-span bridges.Such monitoring systems have pro-vided data that is useful for improvingour understanding of bridge vibrationproblems.179 Furthermore, monitoringdata will be essential for validation ofstructural models and for providing

design guidelines of pedestrian bridgesin several countries.

For the lateral synchronization prob-lem, while the governing mechanismthat generates the load is still some-what debatable and remains a subjectof research, there is one thing that iscommonly agreed upon, that is the

amplitude of lateral vibration and theoccurrence of input force synchro-nization depending on the numberof pedestrians crossing the bridge.Therefore, limiting the number of peo-ple could be one way to avoid excessivelateral vibration. However, for longerspan pedestrian bridge this may beimpractical, hence suppressing vibra-tion by increasing damping is a morereasonable option. Analytical modelscan estimate the amount of dampingrequired to suppress the vibration.169In the case of the cable-stayed pedes-trian bridge near Tokyo, TLDs were

used to control the lateral vibration,while in the Millennium Bridge TMDsand viscous dampers were used.175

Design codes for footbridge have takenthe effect of lateral vibration and syn-chronization into account; the Frenchdesign guideline for footbridges,176 forexample, defines the specific accelera-tion criterion as the transition betweenrandom and synchronization. On thebasis of the results of MillenniumBridge study, the FIB design guide-line for footbridges177 specifies thenecessary damping factor and critical

number of people for prohibiting lat-eral synchronization. More recently,there has been a collective effort byEuropean researchers178 to providea more comprehensive guideline fordesign of pedestrian bridges againsthuman-induced vibration. The guide-line is expec

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Vibration Mechanisms and Controls of Long-Span Bridge -Review

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