Engineering Structures 36 (2012) 228–238

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Engineering Structures

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Pedestrian-induced torsional vibrations of suspended footbridges: Proposaland evaluation of vibration countermeasures

Luca Bruno ⇑, Fiammetta Venuti, Vittorio NascéPolitecnico di Torino, Department of Architecture and Design, Viale Mattioli 39, I-10125 Torino, Italy

a r t i c l e i n f o

Article history:Received 6 July 2011Revised 28 November 2011Accepted 2 December 2011Available online 9 January 2012

Keywords:Suspension footbridgePedestrian dynamic loadVibration countermeasures

0141-0296/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.engstruct.2011.12.012

⇑ Corresponding author. Tel.: +39 011 090 4870; faE-mail address: luca.bruno@polito.it (L. Bruno).

a b s t r a c t

The aim of the paper is to propose and evaluate different design arrangements addressed to mitigate thein-service pedestrian-induced torsional vibrations of lightweight suspended footbridges. All the proposedstructural countermeasures are characterised by the addition of few elements to the original structure,i.e. punctual masses and/or additional cables located in specific sections to reduce vertical and longitu-dinal relative displacements of the suspension cables, in order to preserve the bridge lightness and slen-derness. An application to a test case footbridge is provided. Its structural nonlinear analysis is carried outby means of numerical simulations. On the basis of a preliminary modal analysis, a criterion for the local-ization of the vibration countermeasures along the span is proposed. The effectiveness of the localizationcriterion and of the proposed design arrangements is evaluated by comparing the structural responsesobtained through step-by-step dynamic analyses with and without the countermeasures. The analysisof the results allows the design arrangements to be discussed and the best suited ones to be selected.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The suspended bridge structural type has been widelyemployed from ancient times to nowadays in building longer,lighter and slender footbridge structures with limited resourcesto meet aesthetic and functional demands. Among the variousexamples (Fig. 1), let us recall the historical footbridges built overthe Sesia river in Italy [1], the Rhone river in France or the Hida riv-er in Japan [2], the over 500 rescue bridges built by Toni Ruttimannduring the last 25 years in developing countries [3] or the recentM-bridge in Japan [4].

The dynamic response under service loads is one of the key per-formances of these footbridges, usually characterised by a thinplate-like deck, lightweight suspension cables and very low damp-ing. In particular, their dynamic response is highly sensitiveto the pedestrian added mass, and significantly driven by highermodes, for which resonant conditions with the pedestrian loadsare expected. Besides vertical and lateral vibrations, which arewell-known to affect these kind of structures (e.g. [4]), the extremelylow torsional stiffness of the deck and the small main cable spacing-to-span length ratio make them highly prone to torsional vibrationsinduced by the pedestrian loading.

The structural measures commonly adopted to reduce vibrationson footbridges can be classified according to two different ap-proaches [5]: (i) increasing the damping; (ii) increasing the stiffness.

ll rights reserved.

x: +39 011 090 4999.

Damping is usually increased by means of passive control strategies,e.g. through the installation of tuned mass dampers [6] or slottedbolted connection elements that couple frictional dampers withrods [7]. The use of extra damping devices is very effective, but re-quires constant maintenance. Among the arrangements adoptedon suspension footbridges to increase their deck stiffness, let us re-call the installation of reverse profiled cables in both the vertical andhorizontal planes [9,10], or stiffer handrails [11], or the addition ofinclined stays, connecting the top of the tower with the deck(Fig. 1c). It is worth noting that the last arrangement has also beenwidely adopted in historical long-span suspension bridges, like theBrooklin bridge [8].

To the authors’ knowledge, there is no evidence in literature ofstructural measures belonging to the second category specificallyemployed to mitigate torsional vibrations on suspended foot-bridges. On the contrary, different structural countermeasures havebeen proposed in the case of extra-long and long-span suspendedbridges in order to improve their aeroelastic behaviour face to sin-gle-mode or coupled flutter (e.g. [12–16]). Coherently, the effective-ness of these countermeasures is evaluated via the flutter criticalwind speed at ultimate limit state. Many of these arrangementsare characterised by the addition of few elements to the originalstructure, i.e. punctual masses [13,14] and/or additional cables lo-cated in specific sections [15,16]. Therefore, they are well-suited tobe adopted in lightweight suspended footbridges, where the preser-vation of the aesthetics and slenderness are key objectives. More-over, the low costs implied by these arrangements make themsuitable to be used on a number of medium and short span existing

Fig. 1. (a) Baraggiolo footbridge (1890, Varallo, Italy) [1]. (b) Gaietta footbridge (1914, Millesimo, Italy). (c) St. George footbridge after the renovation in 1998 (Lyon, France,Photocredit Dark Gaybeul�). (d) M-bridge (1999, Maple Valley, Japan) [4].

L. Bruno et al. / Engineering Structures 36 (2012) 228–238 229

footbridges, which suffer vibration problems. In applying thearrangements proposed for long-span bridges to footbridges, someimportant differences between the two structural types and the dy-namic loads which they undergo should be taken into account andproperly evaluated:

– in extra long-span bridges the mass of the suspension cables ispredominant with respect to the total mass of the bridge, whilethe same does not hold on footbridges;

– serviceability and not ultimate limit state mostly affects thedynamic performance of footbridges;

– the relative displacements between the suspension cables dueto flutter in long-span bridges are by several orders of magni-tude higher than the ones induced by pedestrian loading onfootbridges;

– in long-span bridges flutter usually mainly excites the firstmodes of vibration while in suspended footbridges, character-ised by very low natural frequencies, the pedestrian dynamicload usually excites higher modes of vibration.

The scope of the paper is to propose and evaluate differentdesign arrangements – some of which inspired by ones proposedfor long-span suspended bridges – to mitigate the pedestrian-induced torsional vibration of suspended footbridges. Section 2 isdevoted to the description of the proposed arrangements and ofa localization criterion to identify the best position of vibrationcountermeasures along the footbridge span. In Section 3 the pro-posed approach is applied to a test case footbridge and dynamicanalyses are performed in order to compare the peak accelerationobtained without and with the vibration countermeasures. Finally,some conclusions are outlined in Section 4.

2. Proposed approach

2.1. Description of vibration countermeasures

Several vibration countermeasures are proposed with the fol-lowing objectives:

– reduce torsional vibrations under pedestrian dynamic loading,that is, reduce the relative vertical and/or longitudinal displace-ments between the suspension cables;

– preserve the original footbridge aesthetic characterised by itslightness and slenderness;

– guarantee cheap, easy to install and on-site tuneablecountermeasures.

Most of the proposed arrangements are inspired by the onesproposed in literature for long-span suspended bridges facing tothe wind load. In fact, some mechanical and geometrical similari-ties between this structural type and lightweight suspension foot-bridges can be observed [17].

The first three arrangements are characterised by the additionof structural elements in given cross sections of the footbridge:

a1. The first one is based on the addition of eccentric masses ma

(Fig. 2a) on the deck, as already proposed in [12,13]. The roleof the added masses is to increase the deck dead load, inorder to increase in turn the tension of the suspension cablesand their geometric stiffness. This arrangement will be cou-pled with the following a2.

a2. The second one, inspired by the proposal in [15,16], is charac-terised by the addition of a horizontal truss between the sus-pension cables that prevents their relative horizontaldisplacements and diagonal hangers, instead of the verticalones, that connect the suspension cables to the deck(Fig. 2b). The diagonal hangers are arranged in order not tointerfere with the passage of the pedestrians and their preten-sion is given by the added masses on the deck.

a3. the last one is characterised by the substitution of the verti-cal hangers with a rigid frame. Three alternative arrange-ments are analysed: a3a (Fig. 2c) is a p-shaped frame, withvertical columns pinned at both ends and diagonal pre-stressed cables that connect the horizontal beam to thedeck; a3b (Fig. 2d) is characterised by a frame pinned tothe deck transversal beam; a3c (Fig. 2e) is the same as a3b,

Fig. 2. Proposed arrangements a1 (a), a2 (b) and a3 (c)–(d)–(e).

Fig. 4. (a) St. George footbridge over the Saone river, Lyon (France) before therenovation in 1998 (Photocredit Gilbert Lamboley� [24]). (b) Reula footbridge overthe Segre river (1986, Alt Urgell, Spain) [8].

230 L. Bruno et al. / Engineering Structures 36 (2012) 228–238

but columns are fixed at both ends. Arrangements a3 imply astronger impact on the bridge aestethics with respect to theother ones.

Other two arrangements are characterised by the addition ofstructural elements along the span of the footbridge:

a4. The first one (Fig. 3a) is intended to reduce longitudinal rel-ative displacements and is characterised by the addition ofcross stays and trusses connecting the two suspensioncables around the sections where the maximum longitudinalrelative displacement is attended [16].

a5. The second one is based on the addition of inclined staysfrom the footings of the towers to the suspension cables(‘negative stays’ [8]). The arrangement, that has beenadopted in the past on some historical footbridges (Fig. 4a)or in the more recent ones over the river Segre in Spain(Fig. 4b), was mainly conceived to reduce the static deflec-tion. It is expected to be effective to decrease both verticaland longitudinal relative displacements and is alternativeto the more classical one in which the stays connect thetop of the tower to the deck longitudinal girders (‘positivestays’, Fig. 1c).

It is worth pointing out that in long-span suspended bridges thefirst antisymmetric torsional mode is usually the one excited bythe wind load. For this reason, the countermeasures of type a1 toa3 are usually located at one quarter and three quarter of the span[16], where the excited mode shape has its antinodes. In the case ofsuspended footbridges excited by the pedestrian load, the bestlocation of the vibration countermeasures along the span will bedetermined on the basis of the modes which are expected to beexcited and with the criterion described in the next section.

2.2. Description of the localization criterion

This section is devoted to the proposal of a criterion that allowsthe best suited position n 2 ½0; L� along the footbridge span to be

Fig. 3. Proposed arrangements a4 (a) and a5 (b).

selected for the location of the vibration countermeasures. It isworth recalling that the latter are conceived to minimise the rela-tive displacements between the two suspension cables both in thevertical and longitudinal direction. Hence, the best suited spanwiseposition of the measures is expected to be the one at which themaximum relative displacement occurs. The proposed criterion isinspired by these remarks and developed from the results of themode extraction only.

Let us consider the modal equation of motion for mode n:

€qnðtÞ þ 2fn _qnðtÞ þx2nqnðtÞ ¼

FnðtÞMn

; ð1Þ

where qnðtÞ is the principal coordinate, fn is the nth modal dampingratio, xn ¼ 2pfn is the nth natural circular frequency being fn thenatural frequency, FnðtÞ is the modal force and Mn the modal mass.The modal mass and force can be expressed as [18]:

L. Bruno et al. / Engineering Structures 36 (2012) 228–238 231

Mn ¼Z L

0msðxÞ/2

nðxÞdx; ð2Þ

FnðtÞ ¼Z L

0Fðx; tÞ/nðxÞdx;

where L is the length of the footbridge span, /n is the nth unity-scaled mode shape, ms the structural mass per unit length. The timedependent force exerted by a single pedestrians is expressed by thesimplified model

Fðx; tÞ ¼ �F sinðxpv tÞ; ð3Þ

where �F and xpv are the amplitude and circular frequency of thevertical component of the force, respectively.

The contribution of each mode to the structural response isevaluated by means of a weight function:

ctot;n ¼ cðxr;nÞ cðMnÞ; ð4Þ

where the two functions cðxr;nÞ and cðMnÞ are defined as follows.

cðxr;nÞ ¼Dðxr;nÞ

Dðxr;n ¼ 1Þ ¼ Dn2fn; ð5Þ

where xr;n ¼ xpv=xn is the frequency ratio and Dn ¼ ½ð1�x2r;nÞ

2

þð2fnxr;nÞ2��0:5 is the nth dynamic amplification factor. This func-tion weights the contribution of each mode on the basis of itsdistance from the resonance condition (xr;n ¼ 1).

cðMnÞ ¼ 1�Mn

M; ð6Þ

where M is the total mass of the structure. This function weightseach mode on the basis of its modal mass. Indeed, it is evident fromEq. (1) that the larger Mn, the lower the modal (and hence physical)responses and vice versa [19]. For the structural type object of thisstudy, characterised by extreme lightness and sine-like modeshapes, it is expected that cðxr;nÞ plays a more crucial role thancðMnÞ in weighting each mode contribution.

Let us introduce a general localization function KðxÞ conceivedin a way that the higher its value along the span, the more suitedthe corresponding spanwise position to install the vibration coun-termeasures, the more effective the countermeasures. The functionis defined as:

Kk ¼Xj

n¼1

D/0k;n; ð7Þ

where

D/0k;n ¼ ctot;njD/k;nj ¼ ctot;njð/k;n;A � /k;n;BÞj; ð8Þ

with /k;n;A and /k;n;B the kth component (k ¼ x longitudinal, k ¼ yvertical) of the nth mode shape for the suspension cables A and B,respectively. It is worth pointing out that the number j of modesto be retained to evaluate K can be defined according to differentcriteria. For instance, every mode with frequency below a giventhreshold value, the modes with the highest value of ctot or themodes within the maximum risk of resonance according to the clas-sification proposed in [20] could be selected.

It is worth noting that the localisation function can have multi-ple local maxima with close values in ½0; L�, because it results fromthe weighted sum of periodic functions (Eqs. 7, 8). From anengineering point of view, this means that several spanwise loca-tions candidate to host the countermeasures showing analogouseffectiveness. In this case, a two-steps strategy is proposed to se-lect the abscissa n of the countermeasure location among theabscissa of the local maxima argmaxlocðKÞ. The first step involvesa selection of the local maxima based on the localisation functiononly. Local maxima with values close enough to the absolutemaxima are selected by introducing the ratio g:

g ¼maxðKÞ �maxlocðKÞmaxðKÞ ; 0 6 g 6 1; ð9Þ

where maxðKÞ and maxlocðKÞ are the absolute and local maxima of thefunction, respectively. The local maxima for which g is less than athreshold value �g are selected: the lower �g, the lower the numberof selected local maxima, and the closer each other their values. Othernon structural selection criteria can be adopted in the second step, ifmultiple alternative locations are selected in the first step. In thisstudy the selection criterion aims at: (i) minimising the interferencewith the passage of pedestrians (specifically in arrangement a2); (ii)minimising the number of cross stays and trusses in arrangement a4and maximising the slope of the negative stay in arrangement a5. Itfollows that the position closest to the footbridge tower is alwayspreferred among the alternative locations.

3. Application and results

3.1. Description of the test case footbridge

The test case suspended footbridge, symmetrical with respect tothe xy and yz planes, is represented in Fig. 5 and its main geomet-rical properties are summarised in Table 1. Its geometric andmechanical features have been set to be representative of light-weight footbridges (e.g. the ones in Fig. 1): the mass of the deckper spanwise unit length is equal to 106 kg/m. Specifically, thedeck consists of main transversal steel beams (UPN140), whichsupport two secondary longitudinal stringers (UPN100) coveredwith steel gratings. The 10 m-high A-shaped towers are made ofsteel tubes and are pinned at the footing.

The 3D m-dofs discrete model is implemented and analysedusing the Finite Element Method (FEM) through the commercialcode Ansys�. The full description of the adopted numerical tech-niques is out of the scope of this paper. Nevertheless, the main fea-tures of the numerical approach are briefly outlined in thefollowing. Each suspension cable and each stay is modelled by multi-ple truss elements, with tension–compression capabilities, whilethe hangers are resistant to tension only. The deck transversal andlongitudinal beams and the towers are modelled by beam elements,with tension–compression, torsion and bending capabilities. Thenon-structural components (paving, handrails) are modelled by no-dal masses located at the intersection between longitudinal andtransversal beams. The element types and properties are summa-rised in Table 2. A damping ratio f ¼ 0:005 is considered and takeninto account by a classical proportional damping model, being thedamping matrix assumed as a linear combination of the stiffnessand mass matrix (Rayleigh damping, [18]). The towers are pinnedat the footings. The deck abutments are restrained to translationalong the y and z axes, while elastic constraints (with stiffness equalto 1e-5 times the axial stiffness of the deck longitudinal beam, thatis, approximately equal to 20 N/m) are imposed in the longitudinaldirection x in order to fullfill symmetrical conditions.

3.2. Footbridge dynamic behaviour without vibrationcountermeasures

In this section, the dynamic behaviour of the test case foot-bridge in its initial conguration a0, i.e. without the vibration coun-termeasures, is analysed. First, a modal analysis is performed toextract mode shapes and frequencies; hence, the dynamic responseof the footbridge under two different load conditions is calculatedby means of step-by-step dynamic analysis.

3.2.1. Modal analysisDue to the characteristic nonlinearity of suspension bridges, the

modal analysis is performed after a nonlinear static analysis of

Fig. 5. Geometry of the test case suspended footbridge.

Table 1Geometrical data for the test case suspended footbridge.

Main span L ¼ 100 mSag s ¼ 8:5 mHangers spacing d ¼ 2:5 mTowers height H ¼ 10 mDistance between suspension cables 2be ¼ 2:5 mDistance between longitudinal beams 2bi ¼ 1:5 mDeck width 2bd ¼ 1:6 m

Table 2Element properties for the test case suspended footbridge.

A ðm2Þ Jz ðm4Þ Jy ðm4Þ E(GPa)

Mass (kg/m – kg)

Deck trans.beam

0.0020 6:05e�6 6:25e�7 210 16

Deck long. beam 0.0014 2:05e�6 2:92e�7 210 10.6Tower 0.0083 7:15e�5 7:15e�5 210 68.4Cable 0.0013 159 10.4Hanger 0.0002 210 1.6Nodal masses 68

Fig. 6. Frequencies in Hz of the first 80 modes.

232 L. Bruno et al. / Engineering Structures 36 (2012) 228–238

the structure, subjected to dead load and prestress, in order todetermine the geometric tangent stiffness matrix to be used inthe modal analysis. The frequencies of the first 80 modes extractedare plotted versus the mode number in Fig. 6 and classified accord-ing to their predominant component. It can be observed that thereis a significant number of local (cable) modes and that natural fre-quencies are really close to one another, as typically happens inthis kind of structures. The mode frequency axis is mapped accord-ing to the risk of resonance for vertical vibrations as proposed inthe Setra/AFGC guideline [20]. The cumulative number of verticaland torsional modes belonging to each class is graphed in Fig. 7.No torsional modes fall into the maximum risk range, while fourmodes (10, 19, 27, 28) fall in the medium risk range. It is worth

pointing out than two of them (27 and 28) have frequencies thatare close to the boundary value between maximum and mediumrisk. Fig. 8 reports the mode shape and frequencies of the first sixtorsional modes. It is worth pointing out that the first torsionalmode is one-node antisymmetric: this is typical of suspensionbridges with extremely flexible deck, so that the bridge has a modeshape close to the one of the cable alone [21].

3.2.2. Transient analysisNonlinear step-by-step dynamic analyses are performed on the

FE model in order to obtain the maximum accelerations on the foot-bridge in the initial configuration (subscript 0) under a realistic,exploratory service load condition. The load condition is conceivedin order to: (i) compare the effectiveness of the proposed counter-measures under common, exploratory external loads; (ii) induce astructural response characterised by significant contributions frommultiple modes, allowing the effectiveness of the proposed multi-mode localisation criterion to be evaluated; (iii) model a realisticservice load which induces uncomfortable vertical accelerations of

Fig. 7. Classification of the torsional (a) and vertical (b) modes according to the risk of resonance [20].

Fig. 8. Mode shape of the first six torsional modes.

Fig. 9. Scheme of the load conditions.

Table 3

L. Bruno et al. / Engineering Structures 36 (2012) 228–238 233

the deck without countermeasures. Bearing in mind these objec-tives, the simulations are performed by considering the travellingload of a single pedestrian walking at constant speed v [22]:Fðx; tÞ ¼ FðtÞdðxp � vtÞ; ð10Þ

where d is the Dirac operator, xp is the pedestrian position along thebridge span and the time component FðtÞ is expressed according toEq. (3), with �F ¼ 280 N [20], xpv ¼ 4p rad/s (that is, fpv ¼ 2 Hz) andv ¼ 1:5 m=s. The total time of simulation T is set equal to the timeneeded to cross the footbridge length walking at velocity v, that is,T ¼ L=v ¼ 66:67 s. The following load conditions (LC) are consid-ered (Fig. 9):

LCa: one pedestrian walks keeping his/her right, so that he/sheexcites both torsional and vertical motion of the deck;

LCb: two pedestrians walk at a distance equal to the width ofthe walking path in opposition of phase, so that torsional vibra-tions only are induced.

Maximum vertical and lateral acceleration of the deck.

LCa LCb

€ymax;0 ðm=s2Þ 1.7803 2.482x=L 0.175 0.625€zmax;0 ðm=s2Þ 0.0331 0.0756x=L 0.6 0.6

The results of the simulations in terms of maximum value of thevertical and lateral acceleration of the deck are reported in Table 3.Load condition LCb is the one which induces the highest value ofthe structural response, with a maximum vertical acceleration thatfalls close to the boundary between minimum comfort (1

< €y 6 2:5 m=s2) and discomfort classes (€y > 2:5 m=s2), accordingto the classification in [20]. Therefore, the vibration countermea-sures will be tested at LCb only.

In order to evaluate the influence of each mode on the struc-tural response over the simulation time T as a function of the pe-destrian position along the span, a criterion analogous to the

Fig. 10. (a) MAC envelopes versus t=T and (b) mode shapes of the first six modesagainst x=L.

Fig. 12. Weighting functions cðxr;n Þ (a), cðMn Þ (b) and ctot (c).

234 L. Bruno et al. / Engineering Structures 36 (2012) 228–238

Modal Assurance Criterion (MAC) [23] is calculated for the first sixmodes and for LCb:

MACnðtÞ ¼/T

nyðtÞ�� ��2

/Tn/n yTðtÞyðtÞ

; ð11Þ

where /n is the eigenvector of the nth mode and yðtÞ is the vector ofvertical displacements of the footbridge deck. In Fig. 10a the enve-lopes of MACnðtÞ are plotted against the dimensionless time t=T ,while Fig. 10b recalls the eigenvectors against the dimensionlessspace variable x=L. The following considerations can be done: mode28 is the one which mostly contributes to the structural response,since the value of MAC is higher than 0.9 for almost all the timeT; mode 27 is mainly excited when the pedestrian is around thepositions x=L ¼ 0:25 and x=L ¼ 0:85, which correspond to the modeshape antinodes; modes 10 and 19 reach a value of MAC around 0.6when the pedestrian crosses the bridge at x=L ffi 0:1 and x=L ffi 0:5,respectively; the other modes have a minor influence.

3.3. Footbridge dynamic behaviour with vibration countermeasures

In this section, the vibration countermeasures a1 to a5 areapplied to the test case footbridge. First, the localization criterionis applied to find the best location of the countermeasures on thebasis of the modal analysis performed in Section 3.2.1; hence,step-by-step dynamic analyses are performed on the FE model in

Fig. 11. Vertical (a) and longitudinal (b) rela

order to compare the maximum accelerations on the footbridgein the initial configuration (Table 3) with the ones obtained whenthe vibration countermeasures are considered. The simulations areperformed by applying load condition LCb only.

3.3.1. Location of the vibration countermeasuresThe localization criterion described in Section 2.2 is applied to

the test case footbridge. Fig. 11 plots the vertical and longitudinal

tive components of cable mode shapes.

Fig. 13. Vertical (a) and longitudinal (b) weighted relative components of cable mode shapes.

Fig. 14. Localization functions obtained considering different number of modes.

Table 4Location n=L of the vibration countermeasures for �g ¼ 0:3.

All modes Modes 27 28 36 53 Modes 27 28 19 10

Vertical 0.15 0.125 0.15Longitudinal 0.075 0.125 0.15

Table 5Maximum relative vertical and longitudinal displacements of the suspension cablesand their position along the span.

LCa LCb

Dymax;0 ðmÞ 0.0162 0.0317x=L 0.15 0.15Dxmax;0 ðmÞ 0.0037 0.0073x=L 0.125 0.125

L. Bruno et al. / Engineering Structures 36 (2012) 228–238 235

relative components of the first six torsional mode shapes for thesuspension cables.

Fig. 12 graphs the weight functions. As expected, the functioncðxr;nÞ (Fig. 12a) has a more relevant role in weighting the contribu-tion of each mode than cðMnÞ (Fig. 12b). It can be observed that thetwo modes with the highest value of ctot (Fig. 12c) (modes 27 and28) are also the ones with the highest values of MAC (see Fig. 10),that is, the ones which mostly contributes to the bridge dynamicresponse.

Fig. 13 plots the vertical and longitudinal weighted relativecomponents D/0x;n and D/0y;n of the torsional mode shapes for thesuspension cables for the first six modes. The localization functionsplotted in Fig. 14 for half span have been obtained by consideringdifferent number of modes according to the general criteria intro-duced in Section 2.2: all torsional modes with frequency below5 Hz; the first four modes with the highest value of ctot (# 27 2836 53); the modes located in the maximum and medium risk ofresonance according to Setra/AFGC (# 27 28 19 10). It can beobserved that: (i) the function has a similar trend for the threecases, but the position of the maxima slightly differs; (ii) despite

Fig. 15. Values of n=L for vertical (a) and longitudinal (

the considered number of modes, the localization function Ky

shows two local maxima with almost the same value. The bestlocation n=L is finally determined by identifying, among the localmaxima that satisfy the condition g < �g, the one closest to thetower. Fig. 15 shows the values of n=L obtained with increasingnumber of considered modes (ordered for decreasing value ofctot) and with two different values of �g. It is worth recalling thatthe value of �g closest to zero forces the selection of the absolutemaximum of K, whichever the number of considered modes: thiscauses the location n=L to oscillate between the two designsolutions as the cumulative number of modes increases. On thecontrary, the choice of the higher value of �g allows the selectionof the relative maximum nearest to the tower, causing the conver-gence of n=L for increasing cumulative number of modes.Therefore, the results obtained for �g ¼ 0:3 are retained andsummarized in Table 4.

In order to evaluate a posteriori the applied criterion, Table 5 re-ports the maximum relative vertical and longitudinal displacementsof the suspension cables obtained in the initial configuration:

b) localization functions and different values of �g.

Fig. 16. Peak values of vertical (a) and lateral (b) acceleration with arrangements a1 and a2 for different values of 4ma= �Mn .

Fig. 18. a4 (a) and a5 (b): peak values of vertical acceleration for different values of the shortening ratio SR.

Fig. 17. a3: peak values of vertical (a) and lateral (b) acceleration for different values of the shortening ratio.

236 L. Bruno et al. / Engineering Structures 36 (2012) 228–238

Dymax;0 ¼maxjDyj and Dxmax;0 ¼maxjDxj. It can be observed that theposition along the span where the values of Dymax;0 are attained is thesame obtained through the localization criterion when all modes orthe ones according to the Setra/AFGC criterion are selected. On thecontrary, Dxmax;0 is attained at x=L ¼ 0:125, which corresponds tothe position obtained with the localization criterion when the firstfour modes with the highest values of ctot are considered (see Table

4). The position n=L ¼ 0:15 (and its symmetric along the spann=L ¼ 0:85) is retained in the following.

3.3.2. Transient analysisThe maximum vertical accelerations obtained in the initial con-

figuration are compared with the ones attained in the presence ofthe vibration countermeasures a1 to a5 described in Section 2.1

Fig. 19. Arrangement a5: mode shape, mode number and frequency of the first torsional (a)–(b)–(c) and vertical (d)–(e)–(f) modes for different values of the shortening ratioSR in the negative stays.

L. Bruno et al. / Engineering Structures 36 (2012) 228–238 237

and located along the span according to the results obtained in pre-vious section. Specifically, the following properties of the addedstructural elements are adopted:

– the cable elements in arrangements a2 to a5 have the sameproperties as the vertical hangers;

– the horizontal truss in arrangement a2 is a steel profile with cir-cular hollow section (CHS 76.1 � 3.2 mm);

– the frame elements in arrangements a3 are CHS steel profileswith Sections 96 � 7 mm, except for the columns in a3b thatare 96 � 3 mm CHS;

– the horizontal truss in arrangement a4 is a steel profile with cir-cular hollow section (CHS 48.3 � 1.2 mm).

As far as arrangements a1 and a2 are concerned, a sensitivitystudy on their performance to the added mass ma is carried out. Inparticular, the total added mass (¼ 4ma) varies between zero and20% of the average modal mass of the bridge �Mn. The results of thesimulations are summarised in Fig. 16. The arrangement a1 allowsa reduction of the vertical acceleration of 52.6 % with ma ¼ 112 kg.The rate of reduction is higher for lower values of the added massand decreases as the added mass increases. The arrangement a2without added mass produces a significant reduction of the verticalacceleration of about 33%, while the rate of reduction remains al-most constant and low for increasing values of the added mass.For ma higher than 56 kg, arrangement a1 results more effectivethan arrangement a2. Within the explored range of 4ma= �Mn, thevertical accelerations attained with both arrangements are still inthe class of minimum comfort [20], but almost one half of the ini-tial acceleration. The arrangement a1 is also applied in a differentlocation along the bridge span than the one selected through thelocalization criterion, in order to test the effectiveness of the latter.The adopted alternative location is at x ¼ 0:25L (and its symmetri-cal x ¼ 0:75L), that is, in correspondence of the antinodes of thefirst torsional mode, which is the one usually accounted for in

the evaluation of analogous vibration countermeasures for long-span bridges against flutter (see Section 1). In this case the maxi-mum acceleration has a slight decrease, confirming the importanceand effectiveness of the localization criterion for this kind of foot-bridges. Fig. 16b evidences that, while the lateral accelerations€zmax;1 obtained with arrangement a1 are very small, the inclinedhangers in arrangement a2 cause the lateral acceleration €zmax;2 toincrease up to eight times the maximum lateral acceleration€zmax;0 of the deck without countermeasures (€zmax;0 ¼ 0:0756 m=s2,see Table 3), with values that fall in the range of minimum comfort(€z > 0:3 m=s2). This fact confirms the best overall performances ofarrangement a1.

The performances of arrangements a3a, a4 and a5 are evaluatedfor different values of the shortening ratio (SR) of the added cables,that is, the ratio of the relative displacement between the cableanchorages (e.g. imposed by means of a turnbuckle) and the lengthof the cable.

The effects of arrangement a3 are plotted in Fig. 17: the peakacceleration is plotted against the SR on the lower axis, while theupper axis reports the percentage value of the mean cable stressr0 in the dead load configuration with respect to the ultimate cablestress ru ¼ 1450 MPa. Among the three alternative schemes, a3b iscertainly the less effective: indeed, it induces a small decrease(11.27%) of the vertical acceleration, but the lateral accelerationreaches the highest value (0:8 pedestrian=m2), which correspondsto the limit of the discomfort range. With arrangements a3a anda3c, the vertical acceleration significantly decreases (37.8% and51.6%, respectively). The SR in a3a has a minor effect on the struc-tural response, which remains almost constant for values of the SRhigher than 0.76e-3. All a3 arrangements induce peak lateral accel-erations falling in the range of minimum comfort.

Arrangement a4 (Fig. 18a) shows a significant effectiveness inreducing the structural vertical response without inducing largelateral oscillations (0:053 6 €zmax;4 6 0:084), always in the maxi-mum comfort range [20], and almost constant versus SR. The peak

238 L. Bruno et al. / Engineering Structures 36 (2012) 228–238

vertical acceleration in the case of null cable prestress is 30% of theinitial value and further decreases for increasing values of the SR,reaching the range of mean comfort. It should, therefore, be consid-ered that the high tension in the cross stays (up to 35% the ultimatestress) induces high values of compression in the horizontaltrusses, which should be properly resized in a design scenario.

Finally, the peak values of the vertical acceleration obtained witharrangement a5 are plotted in Fig. 18b, while lateral acceleration isnot graphed because its low values (0:023 6 €zmax;5 6 0:073) almostconstant versus SR. Differently from the previous arrangements,the a5 performances do not show a regular and monotonic trend ver-sus the design parameter (i.e. SR or the prestress r0=ru). Thearrangement is highly effective for very low SR value (lower than0.76e-3, i.e. r0=ru < 1:38%), when the vertical acceleration de-creases of around 62% reaching the range of mean comfort. Asthe SR increases, the footbridge gradually undergoes changes inits structural behaviour due to the stress stiffening effects inducedby the negative stays. In particular, the following effects take place:(i) the stays provide an elastic constraint to the suspension cablesso that the suspension cable effective length tends to reduce to thedistance between the stays anchorages L1 (see Fig. 3); (ii) the sag tospan ratio of the suspension cables decreases; (iii) the main cablestiffness increases. As a consequence, the trend of the peak verticalacceleration versus the SR shows two transient regimes character-ised by decreasing branches (‘tuning branches’) followed by anabrupt increase of the acceleration almost up to the initial value€ymax;0 (‘switching branches’), which correspond to a sudden changeof the structural behaviour. This change is confirmed by looking atthe frequencies and mode shapes of the first torsional and verticalmodes for the three values of SR corresponding to the responselocal maxima (highlighted with circles in Fig. 18b). The reductionof the suspension cable effective length is clearly visible in the ver-tical mode starting from the second case (Fig. 19e–f) and in the tor-sional mode in the third case (Fig. 19c). The change in thesuspension cable behaviour also corresponds to a change in thenatural frequency, which is particularly evident in the torsionalmode. It is worth pointing out that this change does not involvethe vertical and torsional modes for the same value of the SR: thisis conjectured to be due to the fact that the stay added flexuralstiffness is higher than the torsional one with respect to the globalflexural and torsional stiffnesses of the footbridge. Once the foot-bridge has reached the new configuration for values of SR higherthan 4.75e-3, the vibration countermeasure shows regular perfor-mances, the trend of the peak acceleration monotonicallydecreases and its value belongs to the mean comfort class for SRvalues higher than 15e-3.

4. Conclusions

In this paper, five structural countermeasures to reduce tor-sional human-induced vibrations on suspended footbridges havebeen proposed and evaluated on a test case footbridge by meansof numerical simulations. All the proposed measures are character-ised by the addition of only few elements to the structure, whilepreserving the footbridge lightness and so allow the applicationon existing footbridges at low costs. The elements added to thestructure are located along the bridge span according to a proposedlocalization criterion based on the minimisation of the relative dis-placements between the suspension cables.

The results of the simulations allow the following consider-ations to be outlined:

– among the different arrangements, considering their overallperformances and drawbacks (e.g. the relevant increase of thelateral acceleration in a2 and a3 or the high compression in

the trusses of a4), a1 seems to be the most effective since itreduces the vertical acceleration by one half with a relativelysmall added mass (20% of the average modal mass). A similarperformance can be obtained with arrangement a5, for suitablevalues of the initial prestress;

– each arrangement should be tuned on the specific footbridge,both to select its best location, which depends on the struc-ture modal properties, and to calibrate the value of the freedesign parameter, that is, the added mass in arrangementsa1 and a2, and the added cable prestress in arrangementsa3a, a4 and a5. Such a fine tuning could be performed bymeans of further numerical simulations or in situ experimen-tal tests on the basis of the general and comparative resultsobtained in the present study. Moreover, the in situ experi-mental tuning can be easily performed because it involvesfew accessible and lightweight structural elements, especiallyin arrangements a1 and a5;

– some of the arrangements could be coupled in order to increasetheir effectiveness (e.g. a4 with a1 or a4 with a2).

References

[1] Re L. Cable suspended: the suspension bridges of the 19th century in Valsesia(in Italian), Lindau, Turin; 1993.

[2] Leonhardt F. Brücken/Bridges: aesthetics and design. Stuttgart: DeutscheVerlags-Anstalt; 1982.

[3] Adev B, Badoux M, Brühwiler E, Ruttimann T. Bridges making a difference.Struct Eng Int 2000;10(1):66–9.

[4] Nakamura S. Field measurement of lateral vibration on a pedestriansuspension bridge. Struct Eng 2003;81(22):22–6.

[5] Zivanovic S, Pavic A, Reynolds P. Vibration serviceability of footbridgesunder human-induced excitation: a literature review. J Sound Vib2005;279:1–74.

[6] Carpineto N, Lacarbonara W, Vestroni F. Mitigation of pedestrian-inducedvibrations in suspension footbridges via multiple tuned mass dampers. J VibControl 2010;16(5):749–76.

[7] Law SS, Wu ZM, Chan SL. Vibration control study of a suspension footbridgeusing hybrid slotted bolted connection elements. Eng Struct 2004;26:107–16.

[8] Troyano LF. Bridge engineering: a global prospective. Cambridge: ThomasTelford; 2003.

[9] Huang M-H. Dynamic characteristics of slender suspension footbridges, Ph.D.thesis, Queensland University of Technology; 2006.

[10] Huang M-H, Thambiratnam DP, Perera NJ. Dynamic performance of slendersuspension footbridges under eccentric walking dynamic loads. J Sound Vib2007;303(1-2):239–54.

[11] Gentile C, Gallino N. Ambient vibration testing and structural evaluation of anhistoric suspension footbridge. Adv Eng Softw 2008;39:356–66.

[12] Diana G, Bruni S, Collina A, Zasso A. Aerodynamic challenges in super long spanbridge design. In: Proceedings international symposium on advances in BridgeAerodynamics; 1998.

[13] Nakazaki S, Yamaguchi H. Preliminary design of super long span suspensionbridges using the temporary additional mass method against a storm. J WindEng Indust Aerodyn 1999;83:317–26.

[14] Phongkumsing S, Wilde K, Fujino Y. Analytical study on flutter suppression byeccentric mass method on fem model of long-span suspension bridge. J WindEng Indust Aerodyn 2001;89:515–34.

[15] Astiz M, Casado C. Flutter stability of very long suspension bridges. J BridgeEng ASCE 1998;3(3):132–9.

[16] Miyazaki M, Arai M, Kazama K, Kubota H. Stay-cable systems of long spansuspension bridges for coupled flutter. In: Proceedings second European andAfrican conference on wind engineering; 1997.

[17] Bruno L, Venuti F, Scotti A. Limit of hanger linearity in suspension footbridgedynamics: a new section model. J Sound Vib 2011. doi:10.1016/j.jsv.2011.07.042.

[18] Clough R, Penzien J. Dynamics of structures. New York: McGraw-Hill; 1987.[19] Brownjohn J, Pavic A. Experimental methods for estimating modal mass in

footbridges using human-induced dynamic excitation. J Sound Vib2007;29(1):2833–43.

[20] Sétra/AFGC, Assessment of vibrational behaviour of footbridges underpedestrian loading (October 2006).

[21] Brownjohn JMW. Vibration characteristics of a suspension footbridge. J SoundVib 1997;202(1):29–46.

[22] Kala J, Salajka V, Hradil P. Footbridge response on single pedestrian inducedvibration analysis. Eng Technol 2009;5:269–80.

[23] Maia NMM, Montalvão e Silva JM. Theoretical and experimental modalanalysis. New York: Research Studies Press Ltd.; 1997.

[24] <http://www.art-ethistoire.com/blc.php?./seguin/pt7lyg.jpg;1340;887./hr/>.