Nonlinear Stochastic Analysis for Lateral Vibration of Footbridge under Pedestrian ...downloads.hindawi.com/journals/mpe/2017/5967491.pdf · · 2017-07-06in 2000 highlight the divergence

Research ArticleNonlinear Stochastic Analysis for Lateral Vibration ofFootbridge under Pedestrian Narrowband Excitation

Jia Bu-yu Yu Xiao-lin Yan Quan-sheng and Yang Zheng

School of Civil Engineering and Transportation South China University of Technology Guangzhou China

Correspondence should be addressed to Yu Xiao-lin xlyu1scuteducn

Received 11 March 2017 Revised 14 May 2017 Accepted 7 June 2017 Published 9 July 2017

Academic Editor Roman Lewandowski

Copyright copy 2017 Jia Bu-yu et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

During the lateral vibration of footbridge the pedestrian lateral load shows two distinct features one is the vibration-dependencyanother is the narrowband randomness caused by the variability between two subsequent walking steps In this case the lateralvibration of footbridge is actually a complicated nonlinear stochastic system In this paper a novel nonlinear stochastic model forlateral vibration of footbridge is proposed in which a velocity-dependent load model developed from Nakamura model is adoptedto represent the pedestrian-bridge interaction and the narrowband stochastic characteristic is consideredThe amplitude and phaseinvolved Ito equations are established using the multiscale method Based on the maximal Lyapunov exponent derived from theseequations the critical condition for triggering a large lateral vibration can be obtained by solving the stability problemThe validityof the proposedmethod is confirmed based on performing the case studies of two bridgesMeanwhile through parameter analysisthe influences of several crucial parameters on the stability of vibration are discussed

1 Introduction

The infamous incidents of large lateral vibrations of theSolferino Bridge in 1999 and the London Millennium Bridgein 2000 highlight the divergence instability in pedestrian-induced vibration that is a slight increase in the numberof people will cause the finite vibration of footbridge todiverge with large amplitude Since these infamous incidentsscholars have begun to study the mechanism of these vibra-tions and gradually realized that the large lateral vibration offootbridge is a complicated process that may be governed byseveral coexisting mechanisms rather than a single one Toexplain such vibration several models have been proposedand classified into deterministic and stochastic models

The representative deterministic models include theFujino [1] Dallard [2] Nakamura [3] Ingolfsson [4 5]Macdonald [6 7] Roberts [8] Newland [9] Piccardo [10]Blekherman [11 12] and Strogatz [13] models The Fujinomodel can be regarded as a linear direct resonance modelwhere the lateral vibrations are caused by direct resonancethat is the pedestrian walking frequency is in resonance withthe natural frequency of one or more lateral vibrationmodesThis model assumes that the vibration amplitude continues

to increase along with the number of pedestrians but suchassumption contradicts the observed sudden divergent vibra-tion on the Millennium Bridge The Dallard Nakamura andIngolfsson models are velocity-dependent nonlinear modelswhere the pedestrian lateral load is assumed to be influencedby the footbridge velocity response and such influence can bedescribed using a knownmodelThe velocity-related terms ina velocity-dependent nonlinear model represent the externaladdition of negative damping to the bridge The criticalnumber of pedestrians needed to trigger the divergence canbe obtained when the overall modal damping becomes neg-ative The Macdonald Roberts and Newland models couplepedestrian motion with bridge vibration as well as consider-ing the influence of footbridge vibration on the pedestrianlateral load However such influence is not described bya known empirical parameter model but rather by cou-pled equations that include pedestrian motion and bridgevibration The Piccardo model is a parametric resonancemodel that attributes the excessive pedestrian-induced lateralvibration in flexible footbridges to a parametric resonancein which the lateral natural frequency is equivalent to halfof the pedestrian lateral walking frequency The Blekhermanmodel is an autoparametric resonance model that assumes

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 5967491 12 pageshttpsdoiorg10115520175967491

2 Mathematical Problems in Engineering

that the vertical and lateral modes are coupled and thatthe vertical excitation energy is transferred to the lateraldirection when the vertical and lateral frequencies of thefootbridge show multiple relationships The Strogatz modelis a pedestrian phase synchronizationmodel that is originallyused to analyze the onset of synchronization in populationsof coupled oscillators According to this model the initiallyrandomly distributed walking frequencies of pedestrians willbe synchronized with that of bridge if an external stimulus(eg vibration amplitude) is strong enough or if the stepfrequency is close to the vibration frequency of the bridge

The major existent models are almost based on deter-ministic methods yet ignore the obvious randomness inpedestrian load Actually the pedestrian load is a complexstochastic process that involves large intrasubject variabilityfrom the same person and intersubject variability amongdifferent people therebymaking the pedestrian load in actualcases become significantly different from that in a determin-istic case This paper focuses on intrasubject variability Inthe deterministic time domain analysis the pedestrian loadis always assumed to be a perfect periodic load that canbe transformed into multiorder harmonic loads by Fourierseries However the real consecutive pedestrian load is not aperfect periodic load but rather a narrowband stochastic pro-cess caused by intrasubject variability If the presented loadis transformed into the frequency domain then the Fourierspectra are not distributed in discrete frequency points as aperfect periodic load but within a certain distributed widtharound the main harmonics thereby leading to a reducedresponse Only few stochastic models exist today such asthe Brownjohn [14] Zivanovic [15] Ingolfsson [4] and Racicmodels [16 17] Although the Brownjohn and Zivanovicmodels aim at the pedestrian vertical load their modelingapproaches for intrasubject randomness can also be appliedto the pedestrian lateral load These two models which canexpress the pedestrian load through their Power SpectralDensity (PSD) allow for the evaluation of the variance of thestructural displacement and acceleration response Howeverthese models do not account for the influence of bridgevibration on pedestrian load The Ingolfsson model aims atthe pedestrian lateral load in which motion-induced forcesincluding equivalent damping and inertia forces are quan-tified through random coefficients that are generated froma discrete-time Gaussian Markov process This model alsoconsiders the randomness of bodyweight walking frequencystep length walking speed and arrival time among differentpeople to represent the intersubject randomness The criticalnumber of pedestrians can be predicted through the criteriaof the initial zero-crossing of overall damping and the firstexceedance of the acceleration threshold although thismodelonly generates numerical results instead of an algebraic solu-tionTheRacicmodel focuses on themultipleGaussian fittingof theAutospectralDensities (ASD) of pedestrian lateral exci-tation by which the variations of amplitude and phase duringthe real pedestrian lateral excitation process can be effectivelyreproduced However this model employs too many empiri-cal parameters to fit numerous Gaussian functions

The tests performed on the Millennium Bridge show thatthe pedestrian lateral load depends on the bridge velocity

In the Dallard model the pedestrian lateral load is thoughtto be proportional to bridge velocity thereby indicating thatthe bridge vibration will reach infinity as the lateral loadincreasesThis result contradicts the fact that pedestrians willeither reduce their walking speed or completely stop whenthe bridge velocity becomes large The Nakamura modelstarts from the Dallard model but introduces a nonlinearvelocity-dependent function to represent the self-limitingnature of pedestrian action (ie the pedestrian response tobridge vibration will saturate under a large bridge velocity)Given that the nonlinear velocity-dependent function adoptsa fraction form and includes an absolute value this modelcannot be used for conveniently implementing an algebraicanalysis Another downside to this model is that the pedes-trian lateral load does not contain harmonic terms therebyindicating that it is independent of walking frequency (iethe effect of the variation of pedestrian walking frequencyon vibration is ignored) Inspired by the Nakamura modelthis paper adopts a nonlinear velocity-dependent model torepresent the pedestrian-bridge interaction but replaces thefraction function with a hyperbolic tangent function like theway that is proposed by Bin and Weiping [18] Meanwhilegiven that the pedestrian gait varies with each step (ie intra-subject variability) the pedestrian lateral load is treated as anarrowband stochastic excitation process to take into accountthe intrasubject variability Unlike the numerical simulationmethod that requires a large number of calculations the pro-posed model is established in a theoretic framework of non-linear stochastic vibration Based on the Ito equations that arederived using the multiscale method the critical conditionfor triggering a large lateral vibration is obtained accordingto the maximal Lyapunov exponent whose sign is used toindicate stability or instability The effectiveness of the pro-posed model is then confirmed through its applications ontheMillenniumBridge (M-Bridge) and the Passerelle Simonede Beauvoir Bridge (P-Bridge) and the influences of somecrucial parameters on the stability of vibration are discussed

2 Model for the LateralVibration of Footbridge

21 Stochastic Model of Pedestrian Lateral Load The Naka-mura model is taken as the basic model to represent thepedestrian lateral load However due to the lack of harmonicterm the Nakamura model can only consider the case ofdirect resonance while it fails to consider the effect of thepedestrian frequency distribution on vibration Obviouslysuch model cannot give a justified explanation for the largelateral vibration of lower frequency bridge modes around05Hz To address this concern a harmonic term (it becomesa stochastic excitation processwhen the intrasubject random-ness is considered more on this later) is introduced into theload model to refine the Nakamura model

Then the lateral load per unit length exerted by pedestri-ans can be defined as follows

119891119888 (119905) = 11989811990111989211988901205880119866 (119910 119910 119910) 120585 (119905) (1)

Mathematical Problems in Engineering 3

minus01 minus005 0 005 01minus1

minus05

0

05

1

tanh model

Fraction function in Nakamura model

G

y (ms)

Figure 1 Relationship between 119866 and 119910 under the hyperbolictangent function and the fraction function in Nakamura model

where119898119901 = 119873119898ps119871with119873 being the number of pedestriansand 119898ps being the single pedestrian mass based on theassumption that the pedestrians are assumed to be uniformlydistributed along the bridge length 119871 119892 is the accelerationof gravity and 1198890 is the dynamic loading factor of the firstharmonic that takes a value of 004 according to [5 14 19](this paper considers only the first harmonic and ignores thecontribution from other high harmonics [20]) 1205880 denotesthe synchronization coefficient that takes a value of 02according to [1 3] 119866(119910 119910 119910) is the vibration-dependentfunction that describes the interaction between pedestrianand footbridge vibration response (ie the displacement119910 velocity 119910 and acceleration 119910) and 120585(119905) denotes thestochastic excitation process (or the harmonic function if thedeterministic periodic load is considered)

As mentioned previously it is not convenient to imple-ment an algebraic analysis for the Nakamura model dueto its assumption that the velocity-dependent function 119866has a fraction form including an absolute value Thereforethe function of 119866 is expressed by the following velocity-dependent hyperbolic tangent function proposed by [18]119866 = 119866 ( 119910) = tanh (119888 119910) (2)

Figure 1 shows the comparison result of 119866 between thehyperbolic tangent function (119888 = 25 sm) and the fractionfunction in the Nakamura model

By employing the Taylor series expansion and ignoringthose terms with orders of larger than three tanh(119888 119910) isrewritten as follows

tanh (119888 119910) = 119888 119910 minus 11988833 1199103 + 119874 ( 1199105) (3)

Substituting (2) and (3) into (1) yields

119891119888 (119905) = 119898119901 (119909) 11989211988901205880 (119888 119910 minus 11988833 1199103)120585 (119905) (4)

Given that intrasubject variability is considered 120585(119905) ischaracterized by a narrowband where PSD obtained from theexperiment is often described by aGaussian-shaped function

The stochastic process with such kind of PSD can bediscretized by using some spectral representation methodsThe method proposed by Shinozuka [21] can be used forthe numerical simulation but this method is not suitablefor algebraic theoretic analysesTherefore an equivalent PSDconversion method is proposed (see Appendix for details)by which the Gaussian-shaped PSD of pedestrian lateralload from existing experiment made by Pizzimenti [22] isconverted into a rational form that is expressed as follows bythe harmonic function with random frequency and phase

120585 (119905) = 120590120585cos (120596119901119905 + 120575119861 (119905) + Φ) (5)

where 120590120585 denotes the excitation intensity 120596119901 denotes thecentral pedestrian walking frequency 120575 denotes the randomdisturbance intensity of frequency 119861(119905) denotes the standardWeiner process and Φ denotes the uniformly distributedphase within the interval of [0 2120587) By employing theequivalent conversionmethod the final equivalent harmonicfunction that represents the pedestrian lateral excitationprocess is obtained as follows

120585 (119905) = 120590120585120585 (119905) = 120590119865ℎ120585 (119905)= 120590119865ℎ cos (120596119901119905 + 120575119861 (119905) + Φ) ℎ = radic 2120587radic2120587119886119904radic2 minus 1198871199042 120575 = radic

2120596119901119887119904radic2 minus 1198871199042 (6)

where 120590119865 corresponds to the quantified coefficient of 120585(119905) in(4) ℎ is regarded as the externally added excitation intensityfactor during the conversion and 120585(119905) denotes the pedestrianlateral stochastic process with unit excitation intensity 119886119904 =09 and 119887119904 = 0043 are the fitting parameters obtained fromthe test results of Pizzimenti and Ricciardelli [22]

22 Equations of Footbridge Lateral Vibration By consideringthe footbridge as an Euler Bernoulli beam its lateral motioncan be expressed as follows

119898119904 (119909) 1205972119910 (119909 119905)1205971199052 + 119888119904 (119909) 120597119910 (119909 119905)120597119905+ 12059721205971199092 [119864119868 (119909) 1205972119910 (119909 119905)1205971199092 ] = 119891119888 (119905) (7)

where 119898119904(119909) 119888119904(119909) and 119864119868(119909) are bridge mass per unitlength bridge damping per unit length and bridge bendingstiffness respectively By assuming that the bridge dampingis proportional (7) can be decoupled into modal differentialequations The first-order mode 119910(119909 119905) = 120593(119909)119902(119905) is onlyconsidered where the mode shape function is consideredas 120601(119909) = sin(120587119909119871) Notably the central span of the M-Bridge (one of the study objects as discussed later) also hasa higher lateral mode second lateral mode with a frequencybeing around 1Hz which could activate a direct resonanceHowever its shape is skew-symmetric and cannot producelarge vibration at midspan unless the pedestrian mass is not


uniformly distributed The corresponding modal differentialequation is expressed as follows

119902 (119905) + 2120596119904120577 119902 (119905) + 1205962119904 119902 (119905) = 1119872119904 int1198710 119891119888 (119905) 120593 (119909) d119909= 119865 (119905) (8)

where 119902(119905) 120596119904 120577 120593(119909)119872119904 and 119865(119905) are the modal displace-ment angular frequency modal damping ratio mode shapemodal mass and mass normalized modal load respectivelyAccording to (4) and (6) the mass normalized modal load119865(119905) is expressed as follows

119865 (119905) = 11988901205880119892119888ℎ119872119904 [int1198710119898119901 (119909) 120593 (119909)2 d119909] 119902120585 (119905)

minus 119889012058801198921198883ℎ3119872119904 [int1198710119898119901 (119909) 120593 (119909)4 d119909] 1199023120585 (119905) (9)

By setting 1198921 = (11988901205880119892119888ℎ119872119904)[int1198710 119898119901(119909)120593(119909)2d119909] and1198922 = (119889012058801198921198883ℎ3119872119904)[int1198710 119898119901(119909)120593(119909)4d119909] (9) becomes

119865 (119905) = 1198921 119902120585 (119905) minus 1198922 1199023120585 (119905) (10)

By substituting (10) into (8) the motion equation offootbridge is expressed as follows

119902 + 2120596119904120577 119902 + 1205962119904 119902 = 1198921 119902120585 (119905) minus 1198922 1199023120585 (119905) (11)

3 Theoretical Derivation Usingthe Multiscale Method

31 Multiscale Method Multiplying the damping term andthe external excitation term in (11) by a small parameter 120576yields

119902 + 2120576120596119904120577 119902 + 1205962119904 119902 = 1205761198921 119902120585 (119905) minus 1205761198922 1199023120585 (119905) (12)

The multiscale method is used to solve (12) The first-order approximate with two time scales is expressed asfollows 119902 (1198790 1198791) = 1199020 (1198790 1198791) + 1205761199021 (1198790 1198791) + sdot sdot sdot (13)

where 1198790 = 119905 and 1198791 = 120576119905 Considering the differentialoperators of 1198630 = 1205971205971198790 and 1198631 = 1205971205971198791 the followingequation is obtained119889119889119905 = 1198630 + 1205761198630 + sdot sdot sdot 11988921198891199052 = 11986320 + 212057611986301198631 + sdot sdot sdot

(14)


11986301199020 + 1205962119904 1199020 = 0 (15)

119863201199021 + 1205962119904 1199021 = minus2119863011986311199020 minus 212059611990412057711986301199020 + 119892111986301199020120585 (119905)minus 1198922 (11986301199020)3 120585 (119905) (16)

The solution to (15) can be written as follows

1199020 (1198790 1198791) = 119860 (1198791) exp (1198941199081199041198790)+ 119860 (1198791) exp (minus1198941199081199041198790) (17)

where 119860(1198791) denotes the complex functions with respect to1198791 and 119860(1198791) denotes the complex conjugate of 119860(1198791)Substituting (6) and (17) into (16) yields

119863201199021 + 1205962119904 1199021 = minus11989421205961199041198601015840 exp (1198941205961199041198790) minus 11989421205962119904 120577119860sdot exp (1198941205961199041198790) + 051198921 119894120596119904119860sdot exp [119894 (120596119901 + 120596119904) 1198790 + 1198941006704120575119861 (1198791) + 119894Φ] minus 119894120596119904119860sdot exp [119894 (120596119901 minus 120596119904) 1198790 + 1198941006704120575119861 (1198791) + 119894Φ]minus 051198922 minus11989412059631199041198603sdot exp [119894 (120596119901 + 3120596119904) 1198790 + 1198941006704120575119861 (1198791) + 119894Φ] minus 11989412059631199041198603sdot exp [minus119894 (120596119901 minus 3120596119904) 1198790 minus 1198941006704120575119861 (1198791) minus 119894Φ]+ 119894312059631199041198602119860 exp [119894 (120596119901 + 120596119904) 1198790 + 1198941006704120575119861 (1198791) + 119894Φ]minus 119894312059631199041198601198602 exp [119894 (120596119901 minus 120596119904) 1198790 + 1198941006704120575119861 (1198791) + 119894Φ]+ cc

(18)

where1198601015840 denotes the derivative with respect to1198791 cc denotesthe complex conjugate of all preceding terms on the right sideof (18) and 1006704120575 = 120575radic120576 because 120575119861(119905) = 120575119861(120576119905)radic120576 = 1006704120575119861(1198791)

On the basis of (18) a parametric resonance will occurwhen 120596119901 asymp 2120596119904 whereas a forced vibration will occurwhen 120596119901 asymp 120596119904 Dallard et alrsquos [23] tests on the LondonMillennium Bridge revealed that the first lateral mode ofthe central span with a frequency of 048Hz was excitedwhich indicates an approximate 1 2 relationship between themode and walking frequency In the normal case except forthe heavy congestion crowd density has a limited influenceon the pedestrian walking frequency and pedestrians areunlikely to slow down their walking frequency to an unusuallevel In this paper a normal walking with a frequency ofapproximately 10Hz (ie approximately twice the walkingfrequency) is assumed and the effect of synchronization isalso considered Given the above consideration it seems thatthe parametric resonance rather than othermechanisms (egthe forced vibration [1] and the nonsynchronization model[4 6]) is reasonable for explaining the large vibration ofthe first lateral mode of the central span of the MillenniumBridge Thus only 120596119901 asymp 2120596119904 is considered in this paperIntroducing a detuning parameter 120576120590 yields the following

120596119901 = 2120596119904 + 120576120590 (19)


Eliminating the secular terms in (18) yields

minus 11989421205961199041198601015840 minus 11989421205962119904 120577119860minus 11989411989212 120596119904119860 exp [1198941205901198791 + 1198941006704120575119861 (1198791) + 119894Φ]+ 11989411989222 12059621199041198603 exp [minus1198941205901198791 minus 1198941006704120575119861 (1198791) minus 119894Φ]+ 119894311989222 12059631199041198601198602 exp [1198941205901198791 + 1198941006704120575119861 (1198791) + 119894Φ] = 0

(20)

By setting 119860 = (12)119886(1198791) exp[119894120593(1198791)] and Δ(1198791) = 1205901198791 +1006704120575119861(1198791) + Φ minus 2120593(1198791) (20) becomes

minus 1198941205961199041198861015840 + 1205961199041198861205931015840 minus 1198941205962119904119886120577+ (1198943119892216 12059631199041198863 minus 11989411989214 120596119904119886) [cosΔ + 119894 sinΔ]+ 11989411989221612059631199041198863 [cosΔ minus 119894 sinΔ] = 0

(21)

Separating (21) into real and imaginary parts yields

1198861015840 = minus120596119904120577119886 minus 11989214 119886 cosΔ + 11989224 12059621199041198863 cosΔ1198861205931015840 = minus11989214 119886 sinΔ + 11989228 12059621199041198863 sinΔ

(22)

As 21198861205931015840 = 119886120590 + 11988610067041205751198611015840(1198791) minus 119886Δ1015840 (22) can be rewritten asfollows

1198861015840 = minus120596119904120577119886 minus 11989214 119886 cosΔ + 11989224 12059621199041198863 cosΔΔ1015840 = (120596119901 minus 2120596119904) + 11989212 sinΔ minus 11989224 12059621199041198862 sinΔ+ 10067041205751198611015840 (119905)

(23)

32 Stability Analysis Implementing a linear operation withregard to the derived coefficients in (23) at 119886 = 0 yields

d119886 = 1d119905 = [minus120596119904120577 minus 1198921 cosΔ4 ] 119886d119905dΔ = 2d119905 + 120575d119861 (119905)= (120596119901 minus 2120596119904 + 1198921 sinΔ2 ) d119905 + 120575d119861 (119905)

(24)

By introducing a new variable Υ = ln 119886 (24) can berewritten as follows

dΥ = (minus120596119904120577 minus 1198921 cosΔ4 ) d119905dΔ = (120596119901 minus 2120596119904 + 1198921 sinΔ2 ) d119905 + 120575d119861 (119905) (25)

Given that Δ(119905) is the ergodic Markov process in theinterval of [0 2120587] its invariant measure represented by the

stationary probability density function 119901(Δ) satisfies thefollowing stationary FokkerndashPlanckndashKolmogorov equation

1198892119901119889Δ2 minus 119889119889Δ [(120590 + 119892 sinΔ) 119901] = 0 (26)

where 120590 = 2(120596119901 minus 2120596119904)1205752 and 119892 = 11989211205752 119901(Δ) satisfies theperiodicity condition 119901(Δ) = 119901(Δ + 2120587) and the probabilitynormalization condition int2120587

0119901(Δ)dΔ = 1 which can be

solved as follows119901 (Δ) = 119862 exp (120590Δ minus 119892 cosΔ)sdot intΔ+2120587Δ

exp (minus120590120591 + 119892 cosΔ) 119889120591 (27)

where119862 is the probability normalization constant Accordingto Oseledecrsquos multiplicative ergodic theorem for any initialvalue (1198860 Δ 0) the Lyapunov exponent 120582(1198860 Δ 0) of thesolution 119886(119905 1198860 Δ 0) of (24) can be defined as follows

120582 (1198860 Δ 0) = lim119905rarrinfin

1119905 ln 1003816100381610038161003816119886 (119905 1198860 Δ 0)1003816100381610038161003816 wp1 (28)

where wp1means with probability one Given that 120582(1198860 Δ 0)only has two different values the stability of the trivialsolution of (24) depends on the maximal Lyapunov exponent120582max The trivial solution of (24) is stable with probabilityone when 120582max lt 0 and unstable with probability onewhen 120582max gt 0 thereby suggesting that the boundarycondition between the stability and instability of (12) can beapproximately represented by 120582max = 0 According to [24]the maximal Lyapunov exponent 120582max can be obtained asfollows by considering the ergodicity of Δ(119905)

120582max = lim119905rarrinfin

1119905 10038161003816100381610038161003816100381610038161003816 119886 (119905)119886 (0) 10038161003816100381610038161003816100381610038161003816 = lim119905rarrinfin

1119905 [Υ (119905) minus Υ (0)]= minus120596119904120577 minus lim

119905rarrinfin

1119905 int1199050 11989214 cosΔ (120591) 119889120591= minus120596119904120577 minus 11989214 119864 [cosΔ]= minus120596119904120577 minus 11989214 int21205870 119901 (Δ) cosΔ119889Δ

(29)

where 119864[lowast] denotes the expected value operator

4 Case Study

41 Critical Number of Pedestrians The central span of theM-Bridge (a shallow suspension bridge located in London)and the central span of the P-Bridge (a combined shallowarch bridge located in Paris) are selected to be the objects ofthe present study According to previous works [2 25 26]the structural parameters of these bridges as well as the othercommon involved parameters are listed in Table 1

According to 120582max = 0 the critical number of pedestrians119873lim (119873 = 119898119901119871119898ps) needed to trigger the unstable lateralvibration of footbridge can be obtained Figure 2 presents


Table 1 Values of parameters

BridgeLength(m)

Firstfrequency

(Hz)

Bridgemass119871(kgm)

Dampingratio ()

Dynamicloadingfactor

Synchronizationcoefficient

Velocityproportionalcoefficient(sm)

Fittingparameters in

PSD

Singlepedestrianmass (kg)

119871 119891119904 119898119904 120577 1198890 1205880 119888 119886119904 119887119904 119898ps

M-Bridge 144 048 2000 07 004 02 25 09 0043 70P-Bridge 190 056 3420 077

092 094 096 098 1 102 104 106 1080

100

200

300

400

500

600

700

800

fr

NＦＣＧ

NＦＣＧ

NＦＣＧ

NＦＣＧ

= 465

= 167

= 79

P-Bridge (ＧＣＨ)

M-Bridge (ＧＣＨ)

P-Bridge (controlled by a metronome) (ＧＣＨ)

Figure 2 Critical number of pedestrians 119873lim as a function of thefrequency ratio 119891119903 for the two bridgesthe critical number of pedestrians as a function of thefrequency ratio between walking frequency and doubledbridge frequency (119891119903 = 1198911199012119891119904) for these two bridgesUnder the worst condition (119891119903 = 1) the minimal valueof 119873lim for the central span of the M-Bridge is 167 whichis consistent with the observations made on the M-Bridge[23] Based on the statements from [25] more than 400people are needed to trigger the instable vibration of the P-Bridge however the controlled walking tests revealed thatan instable vibration with a large amplitude up to 60mmwas reached when a group of 60 people walked in stepusing a metronome to control the walking frequency It isconceivable that the crowd controlled by a metronome willhave a high synchronization and each pedestrian will havea low intrasubject variability Therefore the synchronizationcoefficient is assumed as 08 and the value of 119887119904 used torepresent the intrasubject variability (the role of 119887119904 will bediscussed later) is assumed as half of the normal caseNotablythe assumption lacks the support of available data but itappears at least justifiable in terms of qualitative aspect Thedotted curves in Figure 2 show that the results from theproposedmodelmatchwell with those actual observations onthe P-Bridge The well matched comparative results confirmthe effectiveness of the proposed method

42 Parameter Analysis The frequency distribution numberof pedestrians and random disturbance intensity are taken asthe crucial parameters and their influences on the stability ofthe central span of the M-Bridge are discussed

Figure 3 presents the maximal Lyapunov exponent 120582maxas a function of the number of pedestrians 119873 and thefrequency ratio 119891119903 = 1198911199012119891119904 Based on the mesh surface(Figure 3(a)) 120582max increases along with the number ofpedestrians and reaches its peak when the walking frequencybecomes equivalent to the doubled first lateral frequency(119891119903 = 1) In such case the lateral vibration of bridge willbe in its worst state (lowest stability) To better illustrate theinfluences of119873 and119891119903 on 120582max Figure 3(b) presents the curveof 120582max-119891119903 under different fixed119873 while Figure 3(c) presentsthe curve of 120582max-119873 under different fixed 119891119903 Figure 3(b)shows that when 119873 is relatively small such as 119873 = 100or 119873 = 150 120582max will never exceed zero at any value of119891119903 By contrast when 119873 is large such as 119873 = 200 or119873 = 250 120582max exceeds zero in the area where 119891119901 is closeto 2119891119904 These results reveal that the stability depends on thenumber of pedestrians With a large number of pedestriansthe stability becomes sensitive to frequency distributionHowever with a small number of pedestrians the lateralvibration of the bridge will always remain stable even if thefrequency distribution is within the region of parametricresonance Figure 3(c) shows that the sensitivity of thestability to the number of pedestrians also depends on thefrequency distributionWhen parametric resonance happens(119891119903 = 1) the stability decreases rapidly alongwith the numberof pedestrians When the central lateral walking frequencyis far from the doubled first lateral frequency of the bridge(eg 119891119903 = 11) the stability is only slightly influenced bythe number of pedestrians and the state remains stable evenwhen the number of pedestrians is considerably large

Through the contour of 120582max as shown in Figure 4 theregions of stability and instability are obtained by taking119873 and 119891119903 as the parameters The stability region appearsas a concave shape suggesting that when the number ofpedestrians is less than the value at the lowest point (ie167) the lateral vibration of the bridge will always be stableregardless of how the frequency is distributed which isconsistent with the observations from Figure 3

The influence of the randomness in the pedestrian lateralexcitation on the stability is also considered in the parametricanalysis of the random disturbance intensity of frequency 120575From (6) it can be known that 120575 can be further representedby the parameter 119887119904 (120575monotonically increases with 119887119904)

Figure 5 presents 120582max as a function of 119887119904 and 119891119903 when119873 = 167 Figure 5(a) shows that the peak of 120582max is locatedat the point of minimal 119887119904 and 119891119903 = 1 Figure 5(b) shows thatthe random disturbance intensity also has a great influenceon stability The sensitivity of the stability to the frequency


09095

1105

11

100

250

400520

0

005

01

015

02

N

fr

minus005

ＧＲ

(a)

09 095 1 105 11

0

001

002

003

minus003

minus002

minus001

ＧＲ

fr

N = 250

N = 200

N = 150

N = 100

(b)

100 130 160 190 220 250N

0

001

002

003

minus003

minus002

minus001

ＧＲ

fr = 1

fr = 11

fr = 095

(c)

Figure 3 Maximal Lyapunov exponent 120582max as a function of the number of pedestrians 119873 and the frequency ratio 119891119903 (a) mesh surface of120582max-(119873119891119903) (b) 120582max-119891119903 under fixed119873 and (c) 120582max-119873 under fixed 119891119903

09 095 1 105 11

150

200

250

300

350

minus002

minus001

minus001

minus001

minus001

0

0

0

0 001

001

001

002

002

002

003

003

004004 005

Instability

fr

NＦＣＧ

Figure 4 Contour of 120582max for different values of119873 and 119891119903distribution decreases along with increasing random dis-turbance intensity and the vibration always remains stablewhen the random disturbance intensity is relatively large(eg 119887119904 gt 0083) even if parametric resonance happensSimilar to Figure 3(c) Figure 5(c) shows that the relationshipbetween the stability and the random disturbance intensityis determined by the frequency distribution The stability

increases rapidly along with random disturbance intensitywhen the central lateral walking frequency is close to thedoubled first lateral frequency of the bridge Otherwisethe stability is rarely influenced by the random disturbanceintensity Notably when the frequency is roughly less than095 or larger than 105 the stability decreases slightly alongwith increasing random disturbance intensity

Figure 6 presents the contour of 120582max with respect toparameters 119887119904 and 119891119903 under the case of 119873 = 167 Figure 6shows a small region of instabilityThe lateral vibration of thebridge enters an unstable state only when 0982 lt 119891119903 lt 1017and 119887119904 lt 0043

Figure 7 presents the contours of 120582max with respectto the parameters of 119873 and 119887119904 under the case of 119891119903 =09 (Figure 7(a)) and 119891119903 = 1 (Figure 7(b)) respectivelyThe relationship between 119873 and 119887119904 in the contour line isapproximately linear thereby indicating that the number ofpedestrians has a slight influence on the sensitivity of thestability to the random disturbance intensity and vice versaMeanwhile the stability (or instability) region differs betweenthe cases of 119891119903 = 09 and 119891119903 = 1 The stability tends todecrease along with increasing random disturbance intensityand the number of pedestrians when 119891119903 = 09 but tends toincrease along with increasing random disturbance intensityand decreasing number of pedestrians when 119891119903 = 1 These


09095

1105

11

002004

006008

01

0

001

002

fr

ＧＲ

minus002

minus001

bs

(a)

09 095 1 105 11

0

001

002

ＧＲ

fr

minus003

minus002

minus001

bs = 0023

bs = 0043

bs = 0063

bs = 0083

(b)

003 004 005 006 007 008

0

001

002

ＧＲ

minus003

minus002

minus001

bs

fr = 1

fr = 095

fr = 11

(c)

Figure 5 Maximal Lyapunov exponent 120582max as a function of the number of pedestrians 119887119904 and the frequency ratio 119891119903 (a) mesh surface of120582max-(119887119904 119891119903) (b) 120582max-119891119903 under fixed 119887119904 and (c) 120582max-119887119904 under fixed 119891119903

minus002 minus00

15minus0

015

minus0015

minus0015

minus0015

minus00

15

minus001minus001

minus001

minus00

1

minus001

minus0005

minus0005

minus00

05

0

0 0005

001

Instability

09 095 1 105 11002

003

004

005

006

007

008

fr

b s

Figure 6 Contour of 120582max for different values of 119887119904 and 119891119903findings are consistent with the observations from Figures 3and 5

5 Discussions

This paper proposes a novel theoretic nonlinear stochasticmodel for the lateral vibration of footbridges that adopts a

velocity-dependent hyperbolic tangent function to representthe pedestrian-bridge interaction and considers the nar-rowband stochastic characteristic caused by the intrasubjectvariability By using themultiscalemethod the amplitude andphase involved Ito equations based on the stochastic para-metric resonance mechanism are established by which thecritical condition for triggering the large lateral vibration offootbridges can be obtained by solving the stability problemvia the identification of the sign of the maximal Lyapunovexponent

To the authorsrsquo knowledge the latest published modelsthat can give a justified explanation for the large lateralvibration of lower frequency bridge modes around 05Hzinclude the Piccardo Ingolfsson and Macdonald modelsTo better understand the advantages and limitations of theproposedmodel all of the abovemodels have been comparedas follows

Although the proposed model and the Piccardo modelboth adopt the parametric resonance mechanism to explainthe large lateral vibration in a flexible bridge these modelsshow some distinctive differences For instance the formermodel is stochastic while the latter model is deterministicMoreover the lateral pedestrian-induced force is related to


003 004 005 006 007 008100

200

300

400

500

600

minus002minus002

minus002

minus0015

minus0015

minus0015 minus001

minus001

minus001minus0005

minus0005

minus0005

00

00005

0005Instability

bs

N

(a)

003 004 005 006 007 008100

150

200

250

300

350

minus0015minus0005minus0005

minus0005

0

0

00

0005

00050005

0015

0015

0015004

004

Instability

bs

N

(b)

Figure 7 Contour of 120582max for different values of119873 and 119887119904 (a) 119891119903 = 09 and (b) 119891119903 = 1velocity in the former but is proportional to displacement inthe latterThe comparative results about the stability criterionof these two models show that the proposed model gives amore reasonable result than the PiccardomodelThe stabilitycriterion in the Piccardo model is very sensitive to therelationship between step frequency and modal frequencyA very rapid increase in the critical number of pedestriansis also observed when 119891119901 deviates from 2119891119904 By contrast theresults from the proposed model as shown in Figure 4 seemmore reasonable than those from the Piccardomodel becausethe sensitivity of the critical number of pedestrians to thedistribution of pedestrianwalking frequency has significantlydecreased in the proposed model compared with that in thePiccardo model

Unlike the pedestrian load model presented in thispaper (or the Piccardo model) which is based on full-scalemeasurements the Ingolfsson and Macdonald models arebased on laboratory tests that are related to the latest walkingtests on amoving treadmillThe Ingolfssonmodel is obtainedfrom an extensive experimental campaign on a slightlymodified moving treadmill of Pizzimenti and Ricciardelli[22] The Macdonald model is an inverted pendulum model(IPM) rooted in the field of biomechanics inwhich a steppingcontrol law is adopted based on the instrumented treadmilltest of Hof et al [27] Recently under the IPM frameworkproposed by Macdonald Carroll et al [28 29] rebuilt theexperimental setup developed by Pizzimenti and Ricciardelli[22] and then utilized 3D motion capture equipment to ana-lyze the features of the self-excited force caused by human-structure interaction Similarly Bocian et al [30] conducteda treadmill test based on the IPM framework by using aninteractive virtual reality technology to avoid the implicationsof artificiality and allow for unconstrained gait in the labora-tory environment The main conclusions obtained from thelaboratory tests are as follows the velocity proportional load(or the equivalent negative damping) could be generated eventhough pedestrian lateral walking frequency differs from thatof the bridge (ie synchronization is not necessary for alarge vibration of footbridge) and the large vibration willbe triggered when the equivalent negative damping is equalto the inherent bridge damping In laboratory-based models

(ie the Ingolfsson and Macdonald models) the means ofadopting the velocity proportional load to reflect the effectof structure vibration on pedestrian load and of definingthe stability criterion based on whether total damping isless than zero are similar to those used in this studyAdditionally a certain degree of intrasubject randomnesshas been observed in most of the laboratory tests whichfurther proves the necessity of stochastic analysis The maindifference between the pedestrian load model presented inthis paper and laboratory-based models is that the former isa macroscopic model the parameters of which are estimatedvia back-analysis whereas the latter is a microscopic modelthe parameters ofwhich are obtained directly from laboratorytests Laboratory-based models focus on a single pedestrianbehavior when perturbed by structure motion through sev-eral comprehensive tests which may provide a more precisedescription of the interaction between a single pedestrian anda bridge and can be generalised to other structures Howeverexisting laboratory-based models also exhibit insufficienciesthat are worthy of discussion These models are derivedfrom a test in which only a single pedestrian is walking ona treadmill That is whether these models can be appliedto crowds walking on a bridge is doubtful because thepsychological differences between these two environmentsmay affect a pedestrianrsquos choice of gait and footfall In fact thebehavior of a pedestrian in a crowd considerably differs fromthat when heshe is alone due to the effect of the surroundingcrowd For example in a high density case pedestrians willtend to synchronize to each other due to the attempt to avoidfoot-to-foot and shoulder-to-shoulder contactMoreover theaforementioned laboratory-based models typically involvean elaborate modeling process with a number of variablesand equations which require using a numerical simulationmethod thereby making these models inconvenient to oper-ate Admittedly the pedestrian load model presented in thispaper may be less precise than laboratory-based models fordescribing single pedestrian behavior when walking on amoving platform and thus cannot fully represent the involvedmechanism However given that laboratory-based modelscannot be conveniently used in algebraic analysis in additionto their unconfirmed application to crowd action on a bridge


the pedestrian load model adopted in this study appears atleast justifiable for meeting the requirements of engineeringdesign It is worth emphasizing that the aim of this study isto establish an algebraic framework of nonlinear stochasticvibration for footbridgersquos lateral vibration We believe thatthe application of this framework will not be limited to thecurrent pedestrian load model Nevertheless there are alsosome other limitations of the proposed model This paperconsiders only the first harmonic of pedestrian lateral loadand ignores the higher harmonics Meanwhile for the sakeof simplicity this paper does not consider the intersubjectvariability among crowds and assumes that the synchronizedpedestrians are identical Moreover although the proposedmodel as a kind ofmacroscopicmodel is supported by directevidence on full-scale bridges and is suitable for engineeringapplications its reliability still needs further study becausethe involved parameters are empirical and the amount of con-vincing measurement data remains insufficient Thereforeadditional measurement data and highly precise analyses arerequired to validate this model further

6 Conclusions

Themain conclusions can be summarized as follows

(1) Unlike the numerical simulation method thatrequires a large amount of calculation the proposedmodel is established in an algebraic theoreticframework of nonlinear stochastic vibration

(2) The predicted results based on the proposed modelshow good agreement with the test observations fromthe M-Bridge and the P-Bridge

(3) The frequency distribution significantly influencesthe stability of vibration The worst condition takesplace when the central lateral walking frequency isequal to the doubled first lateral frequency of bridge(ie when parametric resonance happens) In suchcase the vibration stability reaches the lowest levelwhile its sensitivity to the number of pedestrians andthe random disturbance intensity reaches the highestlevel

(4) The number of pedestrians and random disturbanceintensity also greatly influence the stability Increasingthe number of pedestrians will decrease the sta-bility and make the stability more sensitive to thedistribution of frequency The influence of randomdisturbance intensity on the stability depends onthe frequency difference between the pedestrian andbridge When the central lateral walking frequencyapproaches the doubled first lateral frequency ofthe bridge an increase in the random disturbanceintensity will increase the stability and make thestability less sensitive to the distribution of frequencyHowever when the central lateral walking frequencydeviates from the doubled first lateral frequency ofthe bridge the random disturbance intensity hasa relatively small influence on the stability and itsincrease leads to a slight decrease of the stability

Appendix

Equivalent Conversion of PSD

The Gaussian-shaped 119878119865(119891) of the pedestrian lateral excita-tion process as obtained from the test results of Pizzimentiand Ricciardelli [22] takes the following form

119878119865 (119891)1205902119865 = 1119891 119886119904radic2120587119887119904 exp[minus2(119891119891119901 minus 1119887119904 )]2 (A1)

where 1205902119865 = 11988221198892 (119882 is the single pedestrian weightand 119889 is the dynamic loading factor) denotes the doubledarea of PSD around the first load harmonic 119891119901 denotesthe pedestrian walking frequency and 119886119904 = 09 and 119887119904 =0043 are the fitting parameters To facilitate the followingnonlinear stochastic equations the Gaussian-shaped PSDis converted into a rational form that is expressed by theharmonic function Equation (A1) is changed into a functionof angular frequency as follows

119878119865 (120596) = 1205902119865radic2120587120596 119886119904119887119904 exp[minus2(120596120596119901 minus 1119887119904 )]2 (A2)

supporting the fact that the equivalent rational PSD has thefollowing form

119878lowast119865 (120596) = 120590lowast11986522120587 120578(120596119901 minus 120596)2 + 1205782 (A3)

Assuming that the peak values and curvatures of (A2)and (A3) are equal when 120596 = 120596119901 we obtain the following

119878119865 (120596119901) = 119878lowast119865 (120596119901) 11987810158401015840119865 (120596119901) = 11987810158401015840lowast119865 (120596119901) (A4)

According to (A4) we obtain

120578 = 120596119901119887119904radic2 minus 1198871199042 120590lowast1198652 = ℎ21205902119865

(A5)

where ℎ = radic2120587radic2120587119886119904radic2 minus 1198871199042 Based on (A5) (A3) isfurther changed into a two-sided spectrum as follows

119878lowastlowast119865 (120596) = 120590lowast11986522 [119878lowast119865 (120596) + 119878lowast119865 (minus120596)]= ℎ21205902119865 1205782120587 1205962119901 + 1205962 + 1205782(1205962119901 minus 1205962 + 1205782)2 + 412057821205962

(A6)

On the other hand a narrowband process with a rationalspectrum can be generated by a harmonic function with arandom frequency and phase as shown in (5) The PSD of


120585(119905) can be obtained as follows via the Fourier transform ofthe covariance function

119878119865 (120596) = 1205902120585 12057524120587 1205962119901 + 1205962 + 12057544(1205962119901 minus 1205962 + 12057544)2 + 12057541205962 (A7)

By comparing (A6) and (A7) both of these equationsbecome equal when 120575 = radic2120578 and 120590120585 = ℎ120590119865 Finally theequivalent harmonic function for representing the pedestrianlateral excitation process is obtained as shown in (6)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this manuscript

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (nos 51478193 and 51608207) theChinaPostdoctoral Science Foundation (no 2016M592490) theFundamental Research Funds for the Central Universities(no 2015ZM114) and the Open Fund of State Key Laboratoryof Bridge Engineering Structural Dynamics (no 201507)

References

[1] Y Fujino B M Pacheco S-I Nakamura and P WarnitchaildquoSynchronization of human walking observed during lateralvibration of a congested pedestrian bridgerdquo Earthquake Engi-neering and Structural Dynamics vol 22 no 9 pp 741ndash7581993

[2] P Dallard A J Fitzpatrick A Flint et al ldquoThe Londonmillennium footbridgerdquo Structural Engineer vol 79 no 22 pp17ndash33 2001

[3] S-I Nakamura ldquoModel for lateral excitation of footbridges bysynchronous walkingrdquo Journal of Structural Engineering vol130 no 1 pp 32ndash37 2004

[4] E T Ingolfsson and C T Georgakis ldquoA stochastic load modelfor pedestrian-induced lateral forces on footbridgesrdquo Engineer-ing Structures vol 33 no 12 pp 3454ndash3470 2011

[5] E T Ingolfsson C T Georgakis F Ricciardelli and JJonsson ldquoExperimental identification of pedestrian-inducedlateral forces on footbridgesrdquo Journal of Sound and Vibrationvol 330 no 6 pp 1265ndash1284 2011

[6] J H Macdonald ldquoLateral excitation of bridges by balancingpedestriansrdquo Proceedings of The Royal Society of London SeriesAMathematical Physical and Engineering Sciences vol 465 no2104 pp 1055ndash1073 2009

[7] M Bocian J H G MacDonald and J F Burn ldquoBiome-chanically inspired modelling of pedestrian-induced forces onlaterally oscillating structuresrdquo Journal of Sound and Vibrationvol 331 no 16 pp 3914ndash3929 2012

[8] T M Roberts ldquoLateral pedestrian excitation of footbridgesrdquoJournal of Bridge Engineering vol 10 no 1 pp 107ndash112 2005

[9] D E Newland ldquoPedestrian excitation of bridgesrdquo Proceedingsof the Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 218 no 5 pp 477ndash4922004

[10] G Piccardo and F Tubino ldquoParametric resonance of flexiblefootbridges under crowd-induced lateral excitationrdquo Journal ofSound and Vibration vol 311 no 1-2 pp 353ndash371 2008

[11] A N Blekherman ldquoSwaying of pedestrian bridgesrdquo Journal ofBridge Engineering vol 10 no 2 pp 142ndash150 2005

[12] A N Blekherman ldquoAutoparametric resonance in a pedestriansteel arch bridge solferino bridge Parisrdquo Journal of BridgeEngineering vol 12 no 6 pp 669ndash676 2007

[13] S H Strogatz D M Abrams A McRobie B Eckhardt and EOtt ldquoCrowd synchrony on theMillennium BridgerdquoNature vol438 no 7064 pp 43-44 2005

[14] J MW Brownjohn A Pavic and P A Omenzetter ldquoA spectraldensity approach for modelling continuous vertical forces onpedestrian structures due to walkingrdquoCanadian Journal of CivilEngineering vol 31 no 1 pp 65ndash77 2004

[15] S Zivanovic A Pavic and P Reynolds ldquoProbability-basedprediction of multi-mode vibration response to walking exci-tationrdquo Engineering Structures vol 29 no 6 pp 942ndash954 2007

[16] V Racic and J M W Brownjohn ldquoMathematical modelling ofrandom narrow band lateral excitation of footbridges due topedestrians walkingrdquo Computers and Structures vol 90-91 no1 pp 116ndash130 2012

[17] V Racic and J M W Brownjohn ldquoStochastic model of near-periodic vertical loads due to humans walkingrdquo AdvancedEngineering Informatics vol 25 no 2 pp 259ndash275 2011

[18] Z Bin and X Weiping ldquoA nonlinear analysis for the lateralgirder response of footbridges induced by pedestriansrdquo inProceedings of the 2011 International Conference on ElectricTechnology and Civil Engineering ICETCE 2011 pp 4743ndash47462011

[19] X B Yuan Research on Pedestrian-Induced Vibration of Foot-bridge [PhD thesis] Tongji University Shanghai China 2006

[20] A Ebrahimpour A Hamam R L Sack and W N PattenldquoMeasuring and modeling dynamic loads imposed by movingcrowdsrdquo Journal of Structural Engineering vol 122 no 12 pp1468ndash1474 1996

[21] M Shinozuka and G Deodatis ldquoSimulation of stochastic pro-cesses by spectral representationrdquo Applied Mechanics Reviewsvol 44 no 4 pp 191ndash204 1991

[22] A Pizzimenti and F Ricciardelli ldquoExperimental evaluationof the dynamic lateral loading of footbridges by walkingpedestriansrdquo in Proceedings of the 6th international conferenceon structural dynamics Paris France 2005

[23] P Dallard T Fitzpatrick A Flint et al ldquoLondon millenniumbridge pedestrian-induced lateral vibrationrdquo Journal of BridgeEngineering vol 6 no 6 pp 412ndash417 2001

[24] R Hai-wu X Wei W Xiang-dong M Guang and F TongldquoPrincipal response of van der pol-duffing oscillator undercombined deterministic and random parametric excitationrdquoApplied Mathematics and Mechanics vol 23 no 3 pp 299ndash3102002

[25] E T Ingolfsson C T Georgakis and J Jonsson ldquoPedestrian-induced lateral vibrations of footbridges a literature reviewrdquoEngineering Structures vol 45 pp 21ndash52 2012

[26] F Lamarre Passerelle Simone de Beauvoir - Paris 2007Feichtinger Architectes

[27] A L Hof R M van Bockel T Schoppen and K PostemaldquoControl of lateral balance in walking Experimental findings innormal subjects and above-knee amputeesrdquo Gait and Posturevol 25 no 2 pp 250ndash258 2007


[28] S P Carroll J S Owen and M F M Hussein ldquoReproductionof lateral ground reaction forces from visual marker dataand analysis of balance response while walking on a laterallyoscillating deckrdquo Engineering Structures vol 49 pp 1034ndash10472013

[29] S P Carroll J S Owen and M F M Hussein ldquoExperimentalidentification of the lateral human-structure interaction mech-anism and assessment of the inverted-pendulumbiomechanicalmodelrdquo Journal of Sound and Vibration vol 333 no 22 pp5865ndash5884 2014

[30] M Bocian J H G Macdonald J F Burn and D RedmillldquoExperimental identification of the behaviour of and lateralforces from freely-walking pedestrians on laterally oscillatingstructures in a virtual reality environmentrdquo Engineering Struc-tures vol 105 pp 62ndash76 2015

Submit your manuscripts athttpswwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


that the vertical and lateral modes are coupled and thatthe vertical excitation energy is transferred to the lateraldirection when the vertical and lateral frequencies of thefootbridge show multiple relationships The Strogatz modelis a pedestrian phase synchronizationmodel that is originallyused to analyze the onset of synchronization in populationsof coupled oscillators According to this model the initiallyrandomly distributed walking frequencies of pedestrians willbe synchronized with that of bridge if an external stimulus(eg vibration amplitude) is strong enough or if the stepfrequency is close to the vibration frequency of the bridge

The major existent models are almost based on deter-ministic methods yet ignore the obvious randomness inpedestrian load Actually the pedestrian load is a complexstochastic process that involves large intrasubject variabilityfrom the same person and intersubject variability amongdifferent people therebymaking the pedestrian load in actualcases become significantly different from that in a determin-istic case This paper focuses on intrasubject variability Inthe deterministic time domain analysis the pedestrian loadis always assumed to be a perfect periodic load that canbe transformed into multiorder harmonic loads by Fourierseries However the real consecutive pedestrian load is not aperfect periodic load but rather a narrowband stochastic pro-cess caused by intrasubject variability If the presented loadis transformed into the frequency domain then the Fourierspectra are not distributed in discrete frequency points as aperfect periodic load but within a certain distributed widtharound the main harmonics thereby leading to a reducedresponse Only few stochastic models exist today such asthe Brownjohn [14] Zivanovic [15] Ingolfsson [4] and Racicmodels [16 17] Although the Brownjohn and Zivanovicmodels aim at the pedestrian vertical load their modelingapproaches for intrasubject randomness can also be appliedto the pedestrian lateral load These two models which canexpress the pedestrian load through their Power SpectralDensity (PSD) allow for the evaluation of the variance of thestructural displacement and acceleration response Howeverthese models do not account for the influence of bridgevibration on pedestrian load The Ingolfsson model aims atthe pedestrian lateral load in which motion-induced forcesincluding equivalent damping and inertia forces are quan-tified through random coefficients that are generated froma discrete-time Gaussian Markov process This model alsoconsiders the randomness of bodyweight walking frequencystep length walking speed and arrival time among differentpeople to represent the intersubject randomness The criticalnumber of pedestrians can be predicted through the criteriaof the initial zero-crossing of overall damping and the firstexceedance of the acceleration threshold although thismodelonly generates numerical results instead of an algebraic solu-tionTheRacicmodel focuses on themultipleGaussian fittingof theAutospectralDensities (ASD) of pedestrian lateral exci-tation by which the variations of amplitude and phase duringthe real pedestrian lateral excitation process can be effectivelyreproduced However this model employs too many empiri-cal parameters to fit numerous Gaussian functions

The tests performed on the Millennium Bridge show thatthe pedestrian lateral load depends on the bridge velocity

In the Dallard model the pedestrian lateral load is thoughtto be proportional to bridge velocity thereby indicating thatthe bridge vibration will reach infinity as the lateral loadincreasesThis result contradicts the fact that pedestrians willeither reduce their walking speed or completely stop whenthe bridge velocity becomes large The Nakamura modelstarts from the Dallard model but introduces a nonlinearvelocity-dependent function to represent the self-limitingnature of pedestrian action (ie the pedestrian response tobridge vibration will saturate under a large bridge velocity)Given that the nonlinear velocity-dependent function adoptsa fraction form and includes an absolute value this modelcannot be used for conveniently implementing an algebraicanalysis Another downside to this model is that the pedes-trian lateral load does not contain harmonic terms therebyindicating that it is independent of walking frequency (iethe effect of the variation of pedestrian walking frequencyon vibration is ignored) Inspired by the Nakamura modelthis paper adopts a nonlinear velocity-dependent model torepresent the pedestrian-bridge interaction but replaces thefraction function with a hyperbolic tangent function like theway that is proposed by Bin and Weiping [18] Meanwhilegiven that the pedestrian gait varies with each step (ie intra-subject variability) the pedestrian lateral load is treated as anarrowband stochastic excitation process to take into accountthe intrasubject variability Unlike the numerical simulationmethod that requires a large number of calculations the pro-posed model is established in a theoretic framework of non-linear stochastic vibration Based on the Ito equations that arederived using the multiscale method the critical conditionfor triggering a large lateral vibration is obtained accordingto the maximal Lyapunov exponent whose sign is used toindicate stability or instability The effectiveness of the pro-posed model is then confirmed through its applications ontheMillenniumBridge (M-Bridge) and the Passerelle Simonede Beauvoir Bridge (P-Bridge) and the influences of somecrucial parameters on the stability of vibration are discussed

2 Model for the LateralVibration of Footbridge

21 Stochastic Model of Pedestrian Lateral Load The Naka-mura model is taken as the basic model to represent thepedestrian lateral load However due to the lack of harmonicterm the Nakamura model can only consider the case ofdirect resonance while it fails to consider the effect of thepedestrian frequency distribution on vibration Obviouslysuch model cannot give a justified explanation for the largelateral vibration of lower frequency bridge modes around05Hz To address this concern a harmonic term (it becomesa stochastic excitation processwhen the intrasubject random-ness is considered more on this later) is introduced into theload model to refine the Nakamura model

Then the lateral load per unit length exerted by pedestri-ans can be defined as follows

119891119888 (119905) = 11989811990111989211988901205880119866 (119910 119910 119910) 120585 (119905) (1)



minus05

0

05

1

tanh model


G

y (ms)






tanh (119888 119910) = 119888 119910 minus 11988833 1199103 + 119874 ( 1199105) (3)


119891119888 (119905) = 119898119901 (119909) 11989211988901205880 (119888 119910 minus 11988833 1199103)120585 (119905) (4)



120585 (119905) = 120590120585cos (120596119901119905 + 120575119861 (119905) + Φ) (5)



2120596119901119887119904radic2 minus 1198871199042 (6)



119898119904 (119909) 1205972119910 (119909 119905)1205971199052 + 119888119904 (119909) 120597119910 (119909 119905)120597119905+ 12059721205971199092 [119864119868 (119909) 1205972119910 (119909 119905)1205971199092 ] = 119891119888 (119905) (7)




119902 (119905) + 2120596119904120577 119902 (119905) + 1205962119904 119902 (119905) = 1119872119904 int1198710 119891119888 (119905) 120593 (119909) d119909= 119865 (119905) (8)


119865 (119905) = 11988901205880119892119888ℎ119872119904 [int1198710119898119901 (119909) 120593 (119909)2 d119909] 119902120585 (119905)

minus 119889012058801198921198883ℎ3119872119904 [int1198710119898119901 (119909) 120593 (119909)4 d119909] 1199023120585 (119905) (9)


119865 (119905) = 1198921 119902120585 (119905) minus 1198922 1199023120585 (119905) (10)


119902 + 2120596119904120577 119902 + 1205962119904 119902 = 1198921 119902120585 (119905) minus 1198922 1199023120585 (119905) (11)



119902 + 2120576120596119904120577 119902 + 1205962119904 119902 = 1205761198921 119902120585 (119905) minus 1205761198922 1199023120585 (119905) (12)



(14)


11986301199020 + 1205962119904 1199020 = 0 (15)



1199020 (1198790 1198791) = 119860 (1198791) exp (1198941199081199041198790)+ 119860 (1198791) exp (minus1198941199081199041198790) (17)



(18)



120596119901 = 2120596119904 + 120576120590 (19)




(20)



(21)



(22)



(23)



(24)





1198892119901119889Δ2 minus 119889119889Δ [(120590 + 119892 sinΔ) 119901] = 0 (26)




exp (minus120590120591 + 119892 cosΔ) 119889120591 (27)


120582 (1198860 Δ 0) = lim119905rarrinfin

1119905 ln 1003816100381610038161003816119886 (119905 1198860 Δ 0)1003816100381610038161003816 wp1 (28)



1119905 10038161003816100381610038161003816100381610038161003816 119886 (119905)119886 (0) 10038161003816100381610038161003816100381610038161003816 = lim119905rarrinfin


119905rarrinfin


(29)


4 Case Study





BridgeLength(m)

Firstfrequency

(Hz)


Dampingratio ()





PSD


119871 119891119904 119898119904 120577 1198890 1205880 119888 119886119904 119887119904 119898ps

M-Bridge 144 048 2000 07 004 02 25 09 0043 70P-Bridge 190 056 3420 077

092 094 096 098 1 102 104 106 1080

100

200

300

400

500

600

700

800

fr

NＦＣＧ

NＦＣＧ

NＦＣＧ

NＦＣＧ

= 465

= 167

= 79











09095

1105

11

100

250

400520

0

005

01

015

02

N

fr

minus005

ＧＲ

(a)

09 095 1 105 11

0

001

002

003

minus003

minus002

minus001

ＧＲ

fr

N = 250

N = 200

N = 150

N = 100

(b)

100 130 160 190 220 250N

0

001

002

003

minus003

minus002

minus001

ＧＲ

fr = 1

fr = 11

fr = 095

(c)


09 095 1 105 11

150

200

250

300

350

minus002

minus001

minus001

minus001

minus001

0

0

0

0 001

001

001

002

002

002

003

003

004004 005

Instability

fr

NＦＣＧ






09095

1105

11

002004

006008

01

0

001

002

fr

ＧＲ

minus002

minus001

bs

(a)

09 095 1 105 11

0

001

002

ＧＲ

fr

minus003

minus002

minus001

bs = 0023

bs = 0043

bs = 0063

bs = 0083

(b)

003 004 005 006 007 008

0

001

002

ＧＲ

minus003

minus002

minus001

bs

fr = 1

fr = 095

fr = 11

(c)


minus002 minus00

15minus0

015

minus0015

minus0015

minus0015

minus00

15

minus001minus001

minus001

minus00

1

minus001

minus0005

minus0005

minus00

05

0

0 0005

001

Instability

09 095 1 105 11002

003

004

005

006

007

008

fr

b s


5 Discussions






003 004 005 006 007 008100

200

300

400

500

600

minus002minus002

minus002

minus0015

minus0015

minus0015 minus001

minus001

minus001minus0005

minus0005

minus0005

00

00005

0005Instability

bs

N

(a)

003 004 005 006 007 008100

150

200

250

300

350


minus0005

0

0

00

0005

00050005

0015

0015

0015004

004

Instability

bs

N

(b)






6 Conclusions






Appendix









119878119865 (120596119901) = 119878lowast119865 (120596119901) 11987810158401015840119865 (120596119901) = 11987810158401015840lowast119865 (120596119901) (A4)



(A5)



(A6)




119878119865 (120596) = 1205902120585 12057524120587 1205962119901 + 1205962 + 12057544(1205962119901 minus 1205962 + 12057544)2 + 12057541205962 (A7)




Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






minus05

0

05

1

tanh model


G

y (ms)






tanh (119888 119910) = 119888 119910 minus 11988833 1199103 + 119874 ( 1199105) (3)


119891119888 (119905) = 119898119901 (119909) 11989211988901205880 (119888 119910 minus 11988833 1199103)120585 (119905) (4)



120585 (119905) = 120590120585cos (120596119901119905 + 120575119861 (119905) + Φ) (5)



2120596119901119887119904radic2 minus 1198871199042 (6)



119898119904 (119909) 1205972119910 (119909 119905)1205971199052 + 119888119904 (119909) 120597119910 (119909 119905)120597119905+ 12059721205971199092 [119864119868 (119909) 1205972119910 (119909 119905)1205971199092 ] = 119891119888 (119905) (7)




119902 (119905) + 2120596119904120577 119902 (119905) + 1205962119904 119902 (119905) = 1119872119904 int1198710 119891119888 (119905) 120593 (119909) d119909= 119865 (119905) (8)


119865 (119905) = 11988901205880119892119888ℎ119872119904 [int1198710119898119901 (119909) 120593 (119909)2 d119909] 119902120585 (119905)

minus 119889012058801198921198883ℎ3119872119904 [int1198710119898119901 (119909) 120593 (119909)4 d119909] 1199023120585 (119905) (9)


119865 (119905) = 1198921 119902120585 (119905) minus 1198922 1199023120585 (119905) (10)


119902 + 2120596119904120577 119902 + 1205962119904 119902 = 1198921 119902120585 (119905) minus 1198922 1199023120585 (119905) (11)



119902 + 2120576120596119904120577 119902 + 1205962119904 119902 = 1205761198921 119902120585 (119905) minus 1205761198922 1199023120585 (119905) (12)



(14)


11986301199020 + 1205962119904 1199020 = 0 (15)



1199020 (1198790 1198791) = 119860 (1198791) exp (1198941199081199041198790)+ 119860 (1198791) exp (minus1198941199081199041198790) (17)



(18)



120596119901 = 2120596119904 + 120576120590 (19)




(20)



(21)



(22)



(23)



(24)





1198892119901119889Δ2 minus 119889119889Δ [(120590 + 119892 sinΔ) 119901] = 0 (26)




exp (minus120590120591 + 119892 cosΔ) 119889120591 (27)


120582 (1198860 Δ 0) = lim119905rarrinfin

1119905 ln 1003816100381610038161003816119886 (119905 1198860 Δ 0)1003816100381610038161003816 wp1 (28)



1119905 10038161003816100381610038161003816100381610038161003816 119886 (119905)119886 (0) 10038161003816100381610038161003816100381610038161003816 = lim119905rarrinfin


119905rarrinfin


(29)


4 Case Study





BridgeLength(m)

Firstfrequency

(Hz)


Dampingratio ()





PSD


119871 119891119904 119898119904 120577 1198890 1205880 119888 119886119904 119887119904 119898ps

M-Bridge 144 048 2000 07 004 02 25 09 0043 70P-Bridge 190 056 3420 077

092 094 096 098 1 102 104 106 1080

100

200

300

400

500

600

700

800

fr

NＦＣＧ

NＦＣＧ

NＦＣＧ

NＦＣＧ

= 465

= 167

= 79











09095

1105

11

100

250

400520

0

005

01

015

02

N

fr

minus005

ＧＲ

(a)

09 095 1 105 11

0

001

002

003

minus003

minus002

minus001

ＧＲ

fr

N = 250

N = 200

N = 150

N = 100

(b)

100 130 160 190 220 250N

0

001

002

003

minus003

minus002

minus001

ＧＲ

fr = 1

fr = 11

fr = 095

(c)


09 095 1 105 11

150

200

250

300

350

minus002

minus001

minus001

minus001

minus001

0

0

0

0 001

001

001

002

002

002

003

003

004004 005

Instability

fr

NＦＣＧ






09095

1105

11

002004

006008

01

0

001

002

fr

ＧＲ

minus002

minus001

bs

(a)

09 095 1 105 11

0

001

002

ＧＲ

fr

minus003

minus002

minus001

bs = 0023

bs = 0043

bs = 0063

bs = 0083

(b)

003 004 005 006 007 008

0

001

002

ＧＲ

minus003

minus002

minus001

bs

fr = 1

fr = 095

fr = 11

(c)


minus002 minus00

15minus0

015

minus0015

minus0015

minus0015

minus00

15

minus001minus001

minus001

minus00

1

minus001

minus0005

minus0005

minus00

05

0

0 0005

001

Instability

09 095 1 105 11002

003

004

005

006

007

008

fr

b s


5 Discussions






003 004 005 006 007 008100

200

300

400

500

600

minus002minus002

minus002

minus0015

minus0015

minus0015 minus001

minus001

minus001minus0005

minus0005

minus0005

00

00005

0005Instability

bs

N

(a)

003 004 005 006 007 008100

150

200

250

300

350


minus0005

0

0

00

0005

00050005

0015

0015

0015004

004

Instability

bs

N

(b)






6 Conclusions






Appendix









119878119865 (120596119901) = 119878lowast119865 (120596119901) 11987810158401015840119865 (120596119901) = 11987810158401015840lowast119865 (120596119901) (A4)



(A5)



(A6)




119878119865 (120596) = 1205902120585 12057524120587 1205962119901 + 1205962 + 12057544(1205962119901 minus 1205962 + 12057544)2 + 12057541205962 (A7)




Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119902 (119905) + 2120596119904120577 119902 (119905) + 1205962119904 119902 (119905) = 1119872119904 int1198710 119891119888 (119905) 120593 (119909) d119909= 119865 (119905) (8)


119865 (119905) = 11988901205880119892119888ℎ119872119904 [int1198710119898119901 (119909) 120593 (119909)2 d119909] 119902120585 (119905)

minus 119889012058801198921198883ℎ3119872119904 [int1198710119898119901 (119909) 120593 (119909)4 d119909] 1199023120585 (119905) (9)


119865 (119905) = 1198921 119902120585 (119905) minus 1198922 1199023120585 (119905) (10)


119902 + 2120596119904120577 119902 + 1205962119904 119902 = 1198921 119902120585 (119905) minus 1198922 1199023120585 (119905) (11)



119902 + 2120576120596119904120577 119902 + 1205962119904 119902 = 1205761198921 119902120585 (119905) minus 1205761198922 1199023120585 (119905) (12)



(14)


11986301199020 + 1205962119904 1199020 = 0 (15)



1199020 (1198790 1198791) = 119860 (1198791) exp (1198941199081199041198790)+ 119860 (1198791) exp (minus1198941199081199041198790) (17)



(18)



120596119901 = 2120596119904 + 120576120590 (19)




(20)



(21)



(22)



(23)



(24)





1198892119901119889Δ2 minus 119889119889Δ [(120590 + 119892 sinΔ) 119901] = 0 (26)




exp (minus120590120591 + 119892 cosΔ) 119889120591 (27)


120582 (1198860 Δ 0) = lim119905rarrinfin

1119905 ln 1003816100381610038161003816119886 (119905 1198860 Δ 0)1003816100381610038161003816 wp1 (28)



1119905 10038161003816100381610038161003816100381610038161003816 119886 (119905)119886 (0) 10038161003816100381610038161003816100381610038161003816 = lim119905rarrinfin


119905rarrinfin


(29)


4 Case Study





BridgeLength(m)

Firstfrequency

(Hz)


Dampingratio ()





PSD


119871 119891119904 119898119904 120577 1198890 1205880 119888 119886119904 119887119904 119898ps

M-Bridge 144 048 2000 07 004 02 25 09 0043 70P-Bridge 190 056 3420 077

092 094 096 098 1 102 104 106 1080

100

200

300

400

500

600

700

800

fr

NＦＣＧ

NＦＣＧ

NＦＣＧ

NＦＣＧ

= 465

= 167

= 79











09095

1105

11

100

250

400520

0

005

01

015

02

N

fr

minus005

ＧＲ

(a)

09 095 1 105 11

0

001

002

003

minus003

minus002

minus001

ＧＲ

fr

N = 250

N = 200

N = 150

N = 100

(b)

100 130 160 190 220 250N

0

001

002

003

minus003

minus002

minus001

ＧＲ

fr = 1

fr = 11

fr = 095

(c)


09 095 1 105 11

150

200

250

300

350

minus002

minus001

minus001

minus001

minus001

0

0

0

0 001

001

001

002

002

002

003

003

004004 005

Instability

fr

NＦＣＧ






09095

1105

11

002004

006008

01

0

001

002

fr

ＧＲ

minus002

minus001

bs

(a)

09 095 1 105 11

0

001

002

ＧＲ

fr

minus003

minus002

minus001

bs = 0023

bs = 0043

bs = 0063

bs = 0083

(b)

003 004 005 006 007 008

0

001

002

ＧＲ

minus003

minus002

minus001

bs

fr = 1

fr = 095

fr = 11

(c)


minus002 minus00

15minus0

015

minus0015

minus0015

minus0015

minus00

15

minus001minus001

minus001

minus00

1

minus001

minus0005

minus0005

minus00

05

0

0 0005

001

Instability

09 095 1 105 11002

003

004

005

006

007

008

fr

b s


5 Discussions






003 004 005 006 007 008100

200

300

400

500

600

minus002minus002

minus002

minus0015

minus0015

minus0015 minus001

minus001

minus001minus0005

minus0005

minus0005

00

00005

0005Instability

bs

N

(a)

003 004 005 006 007 008100

150

200

250

300

350


minus0005

0

0

00

0005

00050005

0015

0015

0015004

004

Instability

bs

N

(b)






6 Conclusions






Appendix









119878119865 (120596119901) = 119878lowast119865 (120596119901) 11987810158401015840119865 (120596119901) = 11987810158401015840lowast119865 (120596119901) (A4)



(A5)



(A6)




119878119865 (120596) = 1205902120585 12057524120587 1205962119901 + 1205962 + 12057544(1205962119901 minus 1205962 + 12057544)2 + 12057541205962 (A7)




Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







(20)



(21)



(22)



(23)



(24)





1198892119901119889Δ2 minus 119889119889Δ [(120590 + 119892 sinΔ) 119901] = 0 (26)




exp (minus120590120591 + 119892 cosΔ) 119889120591 (27)


120582 (1198860 Δ 0) = lim119905rarrinfin

1119905 ln 1003816100381610038161003816119886 (119905 1198860 Δ 0)1003816100381610038161003816 wp1 (28)



1119905 10038161003816100381610038161003816100381610038161003816 119886 (119905)119886 (0) 10038161003816100381610038161003816100381610038161003816 = lim119905rarrinfin


119905rarrinfin


(29)


4 Case Study





BridgeLength(m)

Firstfrequency

(Hz)


Dampingratio ()





PSD


119871 119891119904 119898119904 120577 1198890 1205880 119888 119886119904 119887119904 119898ps

M-Bridge 144 048 2000 07 004 02 25 09 0043 70P-Bridge 190 056 3420 077

092 094 096 098 1 102 104 106 1080

100

200

300

400

500

600

700

800

fr

NＦＣＧ

NＦＣＧ

NＦＣＧ

NＦＣＧ

= 465

= 167

= 79











09095

1105

11

100

250

400520

0

005

01

015

02

N

fr

minus005

ＧＲ

(a)

09 095 1 105 11

0

001

002

003

minus003

minus002

minus001

ＧＲ

fr

N = 250

N = 200

N = 150

N = 100

(b)

100 130 160 190 220 250N

0

001

002

003

minus003

minus002

minus001

ＧＲ

fr = 1

fr = 11

fr = 095

(c)


09 095 1 105 11

150

200

250

300

350

minus002

minus001

minus001

minus001

minus001

0

0

0

0 001

001

001

002

002

002

003

003

004004 005

Instability

fr

NＦＣＧ






09095

1105

11

002004

006008

01

0

001

002

fr

ＧＲ

minus002

minus001

bs

(a)

09 095 1 105 11

0

001

002

ＧＲ

fr

minus003

minus002

minus001

bs = 0023

bs = 0043

bs = 0063

bs = 0083

(b)

003 004 005 006 007 008

0

001

002

ＧＲ

minus003

minus002

minus001

bs

fr = 1

fr = 095

fr = 11

(c)


minus002 minus00

15minus0

015

minus0015

minus0015

minus0015

minus00

15

minus001minus001

minus001

minus00

1

minus001

minus0005

minus0005

minus00

05

0

0 0005

001

Instability

09 095 1 105 11002

003

004

005

006

007

008

fr

b s


5 Discussions






003 004 005 006 007 008100

200

300

400

500

600

minus002minus002

minus002

minus0015

minus0015

minus0015 minus001

minus001

minus001minus0005

minus0005

minus0005

00

00005

0005Instability

bs

N

(a)

003 004 005 006 007 008100

150

200

250

300

350


minus0005

0

0

00

0005

00050005

0015

0015

0015004

004

Instability

bs

N

(b)






6 Conclusions






Appendix









119878119865 (120596119901) = 119878lowast119865 (120596119901) 11987810158401015840119865 (120596119901) = 11987810158401015840lowast119865 (120596119901) (A4)



(A5)



(A6)




119878119865 (120596) = 1205902120585 12057524120587 1205962119901 + 1205962 + 12057544(1205962119901 minus 1205962 + 12057544)2 + 12057541205962 (A7)




Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






BridgeLength(m)

Firstfrequency

(Hz)


Dampingratio ()





PSD


119871 119891119904 119898119904 120577 1198890 1205880 119888 119886119904 119887119904 119898ps

M-Bridge 144 048 2000 07 004 02 25 09 0043 70P-Bridge 190 056 3420 077

092 094 096 098 1 102 104 106 1080

100

200

300

400

500

600

700

800

fr

NＦＣＧ

NＦＣＧ

NＦＣＧ

NＦＣＧ

= 465

= 167

= 79











09095

1105

11

100

250

400520

0

005

01

015

02

N

fr

minus005

ＧＲ

(a)

09 095 1 105 11

0

001

002

003

minus003

minus002

minus001

ＧＲ

fr

N = 250

N = 200

N = 150

N = 100

(b)

100 130 160 190 220 250N

0

001

002

003

minus003

minus002

minus001

ＧＲ

fr = 1

fr = 11

fr = 095

(c)


09 095 1 105 11

150

200

250

300

350

minus002

minus001

minus001

minus001

minus001

0

0

0

0 001

001

001

002

002

002

003

003

004004 005

Instability

fr

NＦＣＧ






09095

1105

11

002004

006008

01

0

001

002

fr

ＧＲ

minus002

minus001

bs

(a)

09 095 1 105 11

0

001

002

ＧＲ

fr

minus003

minus002

minus001

bs = 0023

bs = 0043

bs = 0063

bs = 0083

(b)

003 004 005 006 007 008

0

001

002

ＧＲ

minus003

minus002

minus001

bs

fr = 1

fr = 095

fr = 11

(c)


minus002 minus00

15minus0

015

minus0015

minus0015

minus0015

minus00

15

minus001minus001

minus001

minus00

1

minus001

minus0005

minus0005

minus00

05

0

0 0005

001

Instability

09 095 1 105 11002

003

004

005

006

007

008

fr

b s


5 Discussions






003 004 005 006 007 008100

200

300

400

500

600

minus002minus002

minus002

minus0015

minus0015

minus0015 minus001

minus001

minus001minus0005

minus0005

minus0005

00

00005

0005Instability

bs

N

(a)

003 004 005 006 007 008100

150

200

250

300

350


minus0005

0

0

00

0005

00050005

0015

0015

0015004

004

Instability

bs

N

(b)






6 Conclusions






Appendix









119878119865 (120596119901) = 119878lowast119865 (120596119901) 11987810158401015840119865 (120596119901) = 11987810158401015840lowast119865 (120596119901) (A4)



(A5)



(A6)




119878119865 (120596) = 1205902120585 12057524120587 1205962119901 + 1205962 + 12057544(1205962119901 minus 1205962 + 12057544)2 + 12057541205962 (A7)




Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





09095

1105

11

100

250

400520

0

005

01

015

02

N

fr

minus005

ＧＲ

(a)

09 095 1 105 11

0

001

002

003

minus003

minus002

minus001

ＧＲ

fr

N = 250

N = 200

N = 150

N = 100

(b)

100 130 160 190 220 250N

0

001

002

003

minus003

minus002

minus001

ＧＲ

fr = 1

fr = 11

fr = 095

(c)


09 095 1 105 11

150

200

250

300

350

minus002

minus001

minus001

minus001

minus001

0

0

0

0 001

001

001

002

002

002

003

003

004004 005

Instability

fr

NＦＣＧ






09095

1105

11

002004

006008

01

0

001

002

fr

ＧＲ

minus002

minus001

bs

(a)

09 095 1 105 11

0

001

002

ＧＲ

fr

minus003

minus002

minus001

bs = 0023

bs = 0043

bs = 0063

bs = 0083

(b)

003 004 005 006 007 008

0

001

002

ＧＲ

minus003

minus002

minus001

bs

fr = 1

fr = 095

fr = 11

(c)


minus002 minus00

15minus0

015

minus0015

minus0015

minus0015

minus00

15

minus001minus001

minus001

minus00

1

minus001

minus0005

minus0005

minus00

05

0

0 0005

001

Instability

09 095 1 105 11002

003

004

005

006

007

008

fr

b s


5 Discussions






003 004 005 006 007 008100

200

300

400

500

600

minus002minus002

minus002

minus0015

minus0015

minus0015 minus001

minus001

minus001minus0005

minus0005

minus0005

00

00005

0005Instability

bs

N

(a)

003 004 005 006 007 008100

150

200

250

300

350


minus0005

0

0

00

0005

00050005

0015

0015

0015004

004

Instability

bs

N

(b)






6 Conclusions






Appendix









119878119865 (120596119901) = 119878lowast119865 (120596119901) 11987810158401015840119865 (120596119901) = 11987810158401015840lowast119865 (120596119901) (A4)



(A5)



(A6)




119878119865 (120596) = 1205902120585 12057524120587 1205962119901 + 1205962 + 12057544(1205962119901 minus 1205962 + 12057544)2 + 12057541205962 (A7)




Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





09095

1105

11

002004

006008

01

0

001

002

fr

ＧＲ

minus002

minus001

bs

(a)

09 095 1 105 11

0

001

002

ＧＲ

fr

minus003

minus002

minus001

bs = 0023

bs = 0043

bs = 0063

bs = 0083

(b)

003 004 005 006 007 008

0

001

002

ＧＲ

minus003

minus002

minus001

bs

fr = 1

fr = 095

fr = 11

(c)


minus002 minus00

15minus0

015

minus0015

minus0015

minus0015

minus00

15

minus001minus001

minus001

minus00

1

minus001

minus0005

minus0005

minus00

05

0

0 0005

001

Instability

09 095 1 105 11002

003

004

005

006

007

008

fr

b s


5 Discussions






003 004 005 006 007 008100

200

300

400

500

600

minus002minus002

minus002

minus0015

minus0015

minus0015 minus001

minus001

minus001minus0005

minus0005

minus0005

00

00005

0005Instability

bs

N

(a)

003 004 005 006 007 008100

150

200

250

300

350


minus0005

0

0

00

0005

00050005

0015

0015

0015004

004

Instability

bs

N

(b)






6 Conclusions






Appendix









119878119865 (120596119901) = 119878lowast119865 (120596119901) 11987810158401015840119865 (120596119901) = 11987810158401015840lowast119865 (120596119901) (A4)



(A5)



(A6)




119878119865 (120596) = 1205902120585 12057524120587 1205962119901 + 1205962 + 12057544(1205962119901 minus 1205962 + 12057544)2 + 12057541205962 (A7)




Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





003 004 005 006 007 008100

200

300

400

500

600

minus002minus002

minus002

minus0015

minus0015

minus0015 minus001

minus001

minus001minus0005

minus0005

minus0005

00

00005

0005Instability

bs

N

(a)

003 004 005 006 007 008100

150

200

250

300

350


minus0005

0

0

00

0005

00050005

0015

0015

0015004

004

Instability

bs

N

(b)






6 Conclusions






Appendix









119878119865 (120596119901) = 119878lowast119865 (120596119901) 11987810158401015840119865 (120596119901) = 11987810158401015840lowast119865 (120596119901) (A4)



(A5)



(A6)




119878119865 (120596) = 1205902120585 12057524120587 1205962119901 + 1205962 + 12057544(1205962119901 minus 1205962 + 12057544)2 + 12057541205962 (A7)




Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






6 Conclusions






Appendix









119878119865 (120596119901) = 119878lowast119865 (120596119901) 11987810158401015840119865 (120596119901) = 11987810158401015840lowast119865 (120596119901) (A4)



(A5)



(A6)




119878119865 (120596) = 1205902120585 12057524120587 1205962119901 + 1205962 + 12057544(1205962119901 minus 1205962 + 12057544)2 + 12057541205962 (A7)




Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119878119865 (120596) = 1205902120585 12057524120587 1205962119901 + 1205962 + 12057544(1205962119901 minus 1205962 + 12057544)2 + 12057541205962 (A7)




Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




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