Footbridge 2005 Second International Conference

Transcription

1 Footbridge 5 Second International Conference VANDAL LOADS AND INDUCED VIBRATIONS ON A FOOTBRIDGE Elsa CAETANO Assistant Professor University of Porto Porto, Portugal Álvaro CUNHA Associate Professor University of Porto Porto, Portugal Carlos MOUTINHO Assistant University of Porto Porto, Portugal Summary The aer describes a study to characterise vertical dynamic loads and induced effects on a footbridge. The study comrehends the develoment of a numerical model to characterise the dynamic behaviour of the footbridge, which is exerimentally validated and used to numerically simulate the resonse induced by grous of edestrians synchronised at critical bridge frequencies. The vandal load associated with a single edestrian is characterised and the resonse is calculated considering the measured load, and comared with the measured resonse to the excitation induced by a single edestrian or a grou with varying dimension. The definition of a general load model for vandal excitation is discussed. Keywords: Footbridge; Vibrations; Vandal Loads; Stress-ribbon bridge. Introduction The roblem of footbridge vibrations induced by edestrians is in general analysed from a oint of view of comfort. This strategy stems from the idea that edestrians are not able to cause damage into a footbridge. Such a conclusion was issued by several authors in the ast, like Wheeler [], whose results were based on the investigation of footbridges, and is also corroborated by the long and rich, in terms of life losses, history of footbridge collases. In effect, most of the reorted failures seem to be related with insufficient carrying caacity, or with damage or insufficient resistance of certain imortant connecting devices, rather than dynamic effects of human loads []. The trend to construct rogressively lighter and slender footbridges, also characterised frequently by very low daming, is resonsible for the roneness of modern footbridges to vibrations. Considering the easiness of resonance excitation for certain structures, it becomes questionable whether it is ossible or not for a grou of ersons to affect the structural integrity of a footbridge. Such a situation is reorted for a footbridge in China in 99, in which a severe structural failure has been rovoked by a grou of students who tried to excite a footbridge for fun, causing the death of 38 ersons [3]. The current aer describes a study develoed to aroriately characterize the otential for structural collase of a stress-ribbon footbridge excited by synchronized grous of vandals. The footbridge, located in the camus of the Faculty of Engineering of Porto, has been extensively investigated in the ast, in order to characterise its dynamic behaviour and the lively resonse to different tyes of dynamic loads. The current study attemts to characterise intentional loads alied to deliberately induce resonance effects, here designated as vandal loads, and the corresonding effects on the behaviour of the footbridge, investigating ossible structural collase conditions. The measured force alied by a single edestrian is characterised and comared with load models defined in the ast. The measured resonse to single and grou excitation is used for back-calculation of grou loads. Comlementary numerical simulations are develoed in order to investigate ossible structural collase conditions. It is observed that, due to defaults in the construction rocedure, the bridge exhibits already a certain degree of damage that is essentially associated with cracks at the joints of recast segments. Although the crack width is exected to rogress, it is concluded that very unlikely can a grou of vandals motivate the collase of the bridge, as the accelerations required

2 Footbridge 5 Second International Conference would be so significant that they would comromise the equilibrium of eole and therefore imly an interrution of the excitation long time before the occurrence of a failure.. Characterisation of the bridge. General descrition The footbridge under analysis integrates the new Camus of the Faculty of Engineering of the University of Porto, roviding a edestrian link between the main buildings and the student s canteen and arking areas (Fig. ). It was designed by ENCIL [] and is characterised as a stress-ribbon footbridge formed by two sans of 8m and 3m. The cross-section is formed by restressing cables embedded in a reinforced concrete aroximately rectangular crosssection with external design dimensions of 3.8mx.5m formed after in-situ casting of concrete over m long recast segments hanging from the tensioned cables. A more comlete descrition of the bridge is resented in [5]. Fig. View of footbridge. Static behaviour Previous exerimental and numerical studies develoed on the bridge [5] illustrated the comlex structural behaviour and the significant deendence uon the construction rocedure followed. Therefore the numerical model was develoed trying to simulate the sequence of construction. Using a nonlinear geometric formulation, a finite element mesh was formed initially by a set of tensioned truss elements that were loaded rogressively, suffering large deformations. Beam elements connecting the same nodes were then activated to simulate the concrete hardening. Linear behaviour was assumed at this stage, considering the tangent stiffness matrix calculated at the deformed configuration. Thorough toograhic and vibration surveys and the udate of the finite element model have rovided imortant conclusions concerning the structural behaviour of the footbridge. It was first observed that the static rofile differs slightly from the redicted at design hase, evidencing a maximum deviation by excess of deformation of around.m close to the middle of the largest san. It was suggested that this rofile might be associated with a thicker cross section than the one defined in the design. That fact was confirmed and it was estimated that the deck mass is higher by % than the corresonding to the design. However it was noticed that the.m excess of deformation could not be entirely due to the increase of deck mass, it was necessary that the initial cable tension was lower than the design value to achieve the current rofile. Iterative calculations led to an initial tension of 3kN instead of 3kN, meaning that imortant loss may have occurred either during or at the end of construction. Finally it was shown that the catenary bridge has an intermediate behaviour between a tensioned cable and a beam structure. Moreover, the validation of the numerical model considering the measured natural frequencies and modal shaes demanded that artial rotations were allowed between successive recast segments, meaning the existence of cracks at those critical sections. This idea is reinforced by the fact that contrary to the design rescritions, no final restressing of the cables was alied at the end of construction, and the joints between recast segments were not sealed. Given the slight thickness of the deck, it is then exected that tensile stresses accumulated at those joints may have originated cracks.

3 Footbridge 5 Second International Conference 3. Numerical Model and Exerimental Validation In order to cature the bridge behaviour in the lateral direction, a three-dimensional model was develoed, simulating again the construction sequence, and emloying a set of 8-node shell elements to discretise the deck and the joints. Two series of arallel truss elements 3.8m distant are used to idealise the four cables. These elements are tensioned considering the initial force of around kn, and submitted to their weight and the weight of the concrete deck, alied rogressively from the abutments to the central ier. After deformation, beam and shell elements are activated connecting the two arallel sets of cable nodes. Beam and shell elements with a length of.99m alternate with beam and shell elements.m long, the later simulating the joints between recast elements. The roerties of these elements were iteratively modified till the natural frequencies and mode shaes aroximated the measured values and a slight modification of the initial cable tension was erformed in order to achieve the measured rofile. Using this rocedure, different reduction factors of area and inertia were obtained with resect to the corresonding intact cross sections. Shell elements of 3.8x.m cross section were used to simulate the region where recast elements exist, and the total area and inertia considered (including beam elements) are defined as.56a and.6i, A and I reresenting the area and inertia of the intact cross section in that region. For the joints, shell elements 3.8x.3m were used, the total area and inertia being.5a j and.8i j, A j and I j reresenting the area and inertia of the corresonding intact cross section. Table Calculated and identified natural frequencies and daming coefficients Mode no. Calculated frequency (Hz) Identified frequency (Hz) Daming (%) Calculated modal mass (ton)* Tye of mode First symmetric (two sans, oosite hase) Second anti-symmetric (L=3m) First symmetric (two sans, in-hase) First lateral (two sans) Second anti-symmetric (L=8m) Second symmetric (L=3m) First torsional (two sans, L=3m dominant) Second symmetric (L=8m) First torsional (two sans, L=8m dominant) Third anti-symmetric (L=3m) * referred to unit modal dislacements at the antinodes F=.Hz F=.5Hz F=.3Hz F=.3Hz Fig. Examles of calculated modal shaes

4 Footbridge 5 Second International Conference Table resumes the resulting natural frequencies and modal shaes, and resents a comarison with the corresonding values identified on the basis of a recent modal survey [6]. The classification of vibration modes as symmetric and antisymmetric is used for easier understanding of the structural behaviour. In ractice, as the structure is not exactly symmetric, this characterisation is only aroximate. Fig. shows the modal configurations associated with the most critical modes from the oint of view of edestrian excitation. The analysis of Table shows that the three-dimensional model is slightly stiffer for the lowest vibration modes (excet the first) and more flexible for the higher order modes. The closeness between natural frequencies, the agreement between calculated and identified modal configurations [5,6], and the reduced mechanical roerties of the elements forming the model, evidence the existence of a considerable level of cracking along the deck, which seems articularly concentrated at the joints between recast segments. As for the ossible contribution of the handrail, it is thought that given the existence of several joints along the footbridge, eventually only the first vibration mode is slightly affected.. Vandal loads The deliberate excitation to resonance of a footbridge results from a first evidence of liveliness and can be achieved in different forms, deending on the characteristics of induced vibration: lateral, vertical bending or torsional. For vertical vibrations, skiing on the sot, also designated as juming, is the easiest way to induce resonance. Another ossibility is bobbing, also designated as bending knees excitation. Contrary to the first case, for this excitation tye both feet of the subject are ermanently in contact with the ground. Figure 3 illustrates the force measured through a set of load cells on a small stiff latform induced by the same subject for the two different situations. The eak amlitude of alied force deends on the frequency of excitation and is generally lower for bending knees excitation than for juming. However, when considering grou excitation, it is observed that the degree of synchronisation amongst edestrians is higher for this situation than for juming, therefore it can haen that higher loads are alied to the bridge using this tye of excitation. Normalised Force (F/G) Bending knees Normalised Force (F/G) Juming- low frequency (a) Fig. 3 Vertical excitation induced by (a) Bending knees excitation; (b) Juming. (b). Single edestrian load model The half-sine model defined by Bachmann [7] for jogging and skiing in the range.8hz-3.hz is defined by k F (t ) = π t G sin( ) t t t t < t T () where k is designated as the dynamic imact factor and defined as k = F max G, G is the weight of the edestrian, tyically defined as 8N, t is the contact duration of the feet with the ground, and T = f is the juming eriod. In this model, the imact factor is defined by k t = π ( ) () T as a result of the condition of constant otential energy. Figure (a) reresents the idealised force suerimosed with the

5 Footbridge 5 Second International Conference measured force associated with an average juming frequency of.7hz. The quantities k, t and T are derived based resectively on the average of eak force, contact duration (determined by the time interval with load amlitude lower than % the average eak force), and on the FFT frequency associated with the first harmonic eak of the load record. Figure (b) reresents the variation with t T of the imact factor defined by () as well as the measured values based on a series of juming tests at different ste frequencies. Table systematises the characteristics of measured forces exressed in terms of the contribution in the first three harmonics, average imact factor and contact duration, and average ratio t T. It is confirmed that the contact duration with the ground is higher than.5s, as secified by Bachmann [7]. This duration increases with the decrease of frequency and reresents more than 6% of the ste eriod for all cases, much higher than the values of 5%-33% secified in BS6399 [8] for normal and high juming. Normalised Force (F/G) g g q y Measured Synthesized Theoretical Rigid base Bridge deck t /T (a) (b) Fig. Half-sine load model for juming: (a) measured and synthesized force for juming frequency of.7hz; (b) Imact factor, load model and measured. Imact factor k Table Characteristics of measured force Tye of Contribution in st Contribution in Contribution in 3 rd Imact factor t t/t excitation harmonic nd harmonic harmonic k Bending knees.6hz (.68).88Hz (.).39Hz (.). Juming high.69hz (.98) 5.37Hz (.3) 8.Hz (.6) Juming high.9hz (.).98Hz (.8) 7.7Hz (.) Juming low.95hz (.6) 3.9Hz (.) 5.8Hz (.6) Juming low.9hz (.7) 3.9Hz (.6) 5.76Hz (.) Juming low 3.7Hz (.6) 3.Hz (.) Juming low.6hz (.).98Hz (.6).9Hz (.) Contact duration t.7 Rigid base.6 Bridge deck.5 Ref. [9] f =/T (a) (b) Fig. 5 (a) Measurement of juming load on a load cell installed at an antinode of a critical mode; (b) Variation of contact duration with frequency of juming for measurements erformed on a rigid latform, on the bridge deck and on rigid surface by [9]

6 Footbridge 5 Second International Conference It is questioned whether a vibrating latform can change the imact factor k as a consequence of the increase of the contact duration between the feet and the ground. This situation could be critical at resonance and would result in a lower value of k. Figure (b) reresents some results of load measurements erformed with the set of load cells and rigid latform installed on the bridge deck at the antinodes of the vibration modes and 3 (see Table and Figure 5). The juming frequency was measured in the range.5hz-.9hz, slightly away from resonance, and the eak acceleration recorded at the load cell location was in the order of.3m/s, corresonding to a dynamic dislacement of about mm. For this range of dislacements no significant variation in the value of k was observed, and the contact duration between the feet and ground, reresented in Figure 5(b) does not show a clear increase, meaning that eventually only for situations of resonance with high amlitude of vibration can that duration increase. The listed values are in good agreement with the results of a more comlete study [9] develoed on a rigid/ movable latform involving a grou of 6 to sort students erforming different activities, in which the following arameters have been identified: T =.3±.3; t =.9±.7; k =3.±.5. Figure 5(b) includes some values from this reference, corresonding to equivalent tests develoed at different natural frequencies. Also as a result of this research, it is observed that a descrition of the juming load based on the Hanning function rovides a much better aroximation to a tyical imulse load than the half-sine model. F (t) is then defined as k F (t ) = π t G.5.5 cos ( ) t t < t T t t Figure 6 (a) shows the aroximation by the half-sine and the Hanning models of a tyical measured imulse function associated with a juming frequency of.7hz, evidencing the better quality of the fit. Using this load model, the imact factor k can be re-calculated as t k = ( ) () T Exression () is reresented in Figure 6(b) and comared both with the one resulting from the half-sine model and with exerimental values above referred. It can be concluded that, excet for low juming frequencies (.6Hz and.7hz), the measured values of k fit better curve (). It should be noted that for those lower frequencies more irregular imulses are measured, which exhibit two close eaks, evidencing a two-ste contact of the feet with the ground. (3) Normalised Force (F/G) 3.5 Measured Synth., half-sine 3 Synth., Hanning.5 Sine model Imact factor k Half-sine Rigid base Bridge deck Hanning t /T (a) (b) Fig. 6 (a) Aroximation of measured imulse load by half-sine/ Hanning/ sine models; (b) Imact factor: load models vs exerimental Figure 6(a) also includes the reresentation of a simler load model that is used by some authors, corresonding to a continuous sinusoidal descrition of the imulse load whose dynamic comonent is given by π t F (t ) =.56 G sin ( ) (5) T

7 Footbridge 5 Second International Conference This simle load model rovides a rough aroximation of the measured imulse load. However, the corresonding amlitude is higher than the amlitude in the first harmonic of the imulse load function for the range -3.5 Hz. Therefore, it is exected that this function rovides a conservative estimate of the imulse load associated with each bridge critical frequency. In order to understand the effect on the bridge resonse of the different descritions of the imulse load, a single degree of freedom system was modelled (with natural frequency of.7hz, daming ratio of % and modal mass of 3ton), whose resonse to a continuous juming in resonance was calculated, based on the measured load record, as well as on the synthesized records according to the half-sine and Hanning load models, and on the simle sine model described by (5). The acceleration records reresented in Figure 7 show that due to some scatter in the alied load, the resonse calculated based on the measured imulse record is slightly lower than the resonse calculated using the fitted load models and is very similar to the one obtained based on the sine load model. Load record Peak dis. (mm) Peak acce. (m/s ) Measured 3.7. Fitted Half-sine.5.5 Fitted Hanning.. Sine 3.8. Fig. 7 SDOF resonse to imulse load at resonance considering different load models Acceleration (m/s ) Half-sine Hanning Sine Measured Grou of edestrians load model The definition of a load model associated with a grou of edestrians requires the secification of the grou dimension and of the ossible degree of correlation between the various elements. These two critical arameters deend both on the bridge behaviour and on the corresonding dimensions and utilization. A search in the literature shows a wide variation of roosals. Grundmann et al. [] define an exected maximum acceleration of.7.8g. Bachmann et al. [] roose the consideration of 3 fully synchronised juming edestrians, and a recent research by the RWTH Aachen and by SBP Stuttgart in the context of the Euroean Project SYNPEX [], based on data collected in several lively footbridges, oints to the definition of an equivalent sinusoidal modal force F n (t ) with a linear amlitude defined as a function of the number of edestrians in the grou N, which is extracted based on the known modal mass M n, the maximum acceleration a and estimated daming ξ n of the vibration mode n under evaluation max, n ( C N + C ) sin ( π f t ) F (t ) = a M ξ sin ( π f t ) = (6) n max,n n n n The simle formula defined above incororates the synchronisation effects, as well as the variation of daming with the amlitude of vibration. The two effects have been investigated searately within the current study. Table 3 and Figure 8 systematise measured eak vertical accelerations on various footbridges, as a function of the number of juming edestrians. A first conclusion that can be taken is that extremely high accelerations can be attained, the maximum known corresonding to about g, occurring in a 5m susension footbridge in Poland tested by the Wroclaw University under the action of 9 juming edestrians. Considering in articular the FEUP footbridge, it can be noticed that the acceleration grows to a maximum limit of around g that is reached with a total number of 3 edestrians. Therefore, a balance seems to be reached that allows for an estimation of an equivalent number of edestrians necessary to roduce this maximum acceleration on a single degree of freedom system, or else the equivalent sine force for analysis. Using the finite element model described in Section, the resonse to the measured imulse load was calculated and comared to the recorded acceleration, based both on the daming ratios identified from the ambient vibration tests n

8 Footbridge 5 Second International Conference dislayed in Table and on the corresonding values identified from high amlitude vibration tests. The analysis of Figure 9, showing the three obtained records at mid-san evidences that the stationary comonent of the resonse is almost indeendent from the daming and also that the measured resonse is much lower than calculated. These results seem to be associated with the slight differences between numerical and identified natural frequencies, conjugated with the fact that the imulse load record shows a dominant frequency of.5hz, that is closer to the numerical natural frequency excited at mid-san (.5Hz) than to corresonding exerimental value (.Hz). Table 3 Peak vertical acceleration associated with vandal excitation by grous of edestrians Number of Deutsches Ruit Eutinger Waagsteg edestrians Museum FEUP Wroclaw Freq (Hz) ξ (%) Vertical eak acceleration (m/s ) No. of ersons Ruit Eut. Waags., f Eut. Waagst., f Eut. Waags., f3 FEUP Deutsches Museum Wroclaw Fig. 8 Variation of eak vertical acceleration with number of edestrians associated with juming of a grou The simulation of the resonance condition is therefore develoed by a rescaling of the time axis associated with the recorded imulse load and the calculated resonse is comared to the measured resonse at /3 rd san for a resonance condition in the second mode of vibration (Fig. ), leading to very close resonance accelerations, of the order of.m/s for erson juming. Using this value as reference and comaring to the measured acceleration induced by a varying

9 Footbridge 5 Second International Conference number of edestrians (Fig. 8), an equivalent number of edestrians can be calculated, which is dislayed in Figure. Therefore, based on the current study, the modelling of a grou with 3 juming edestrians could be done based on the consideration of the inut load associated with 6 erfectly synchronised edestrians, emloying the initial daming table for small vibrations. It is noted however that the eak resonant resonse calculated based on the large daming measurements is.5% the eak resonse calculated with the initial daming table. A correction accounting for the daming variation should therefore be introduced, which leads to the equivalent number of 3. for the grou of 3 edestrians. It is of interest to notice that the ensemble of equivalent number of edestrians corrected for the identified daming ratios are limited by the straight line reresented in Figure, defined by N eq =.835 N (7) where N reresents the number of edestrians in the grou. This line was fitted to the results resented in ref. [9] concerning the measured resonse of grous of edestrians juming on a rigid latform. Further validation of this formula is however required x=.5% x=.8% Measured Acceleration (m/s ) Fig. 9 Calculated vs measured acceleration at mid-san induced by the juming of one erson at that location.5 Measured.5 x=.5% x=.8% Acceleration (m/s ) Acceleration (m/s ) Fig. Calculated and measured acceleration at /3 rd san induced by resonant juming of one erson at that location ( nd mode) Equiv. no. synchronised edestrians Exerimental Based on Ref. [9] Exerimental, dam. correct Number of edestrians Fig. Equivalent number of edestrians: FEUP footbridge results and results from [9]

10 Footbridge 5 Second International Conference 5. Conclusions The aer resents an exerimental and numerical study centred in the definition of load models for vandal excitation on a stress-ribbon footbridge. Recent imulse load measurements for vertical juming by a single edestrian have shown to be in agreement with the Bachmann load model. A more recise descrition based on a Hanning function is however resented, which rovides imulse factors better correlated with measurements than the well-known half-sine model. It is shown however that for ractical alications the sine reresentation rovides very close estimates of the bridge resonse to the one measured on the current footbridge. Considering the load model for grou excitation, ractical examles have shown that very high amlitudes of acceleration can be attained. The measurement of the resonant resonse based on a varying number of juming edestrians is used to extract an equivalent number of synchronised edestrians, imlying the correction by the ratio between daming factors for low and high amlitude vibrations. Finally it is of interest to confirm that even for very large accelerations, with eak value of 8.6m/s, the measured dynamic dislacements of 53mm remain much smaller than the static dislacements evaluated for the crowd load of kn/m (.86m), meaning that the stresses associated with vandal loads remain inferior to those associated with static loads, as frequently referred in the literature. Acknowledgements The authors are grateful for the collaboration of FEUP students in the erformance of the dynamic tests. They thank the data from the Wroclaw footbridge rovided by Jaroslaw Zwolski, from Wroclaw University. The suort rovided by the Euroean Commission in the context of the Euroean Research Project Advanced Load Models for Synchronous Pedestrian Excitation and Otimized Design Guidelines for Steel Footbridges (SYNPEX) is fully acknowledged. References [] WHEELER, J., Prediction and control of edestrian induced vibration in footbridges, J. Structural Division, ASCE, Vol. 8, No. ST9,. 5-65, 98. [] WOLMUTH, B. and SURTEES, J., Crowd-related Failure of Bridges, ICE, New Civil Engineering International, October 3, [3] KASPERSKI, M., The resonse of edestrians to lively footbridges, Proc. of the 6 th International Conference on Structural Dynamics, EURODYN 5, Paris, France, Vol., , 5. [] ENCIL, U.P. Pólo, Passagem ara Peões- Projecto Base de Fundações e Estruturas, 998. [5] CAETANO, E. and CUNHA, A., Exerimental and Numerical Assessment of the Dynamic Behaviour of a Stress-ribbon Footbridge, Structural Concrete, Journal of FIB, Vol. 5, No., [6] CUNHA, A., CAETANO, E., MOUTINHO, C. and MAGALHÃES, F., Daming Identification in a Stress-ribbon Footbridge, Proceedings of EURODYN 5, Paris, Setember 5. [7] BACHMANN, H. et al., Vibration Problems in Structures: Practical Guidelines, Birkhäuser Verlag, Basel, 995, 3. [8] BSI, BS6399 Part : Loads for Buildings- Code of Practice for Dead and Imosed Loads. London; UK: British Standards Institution. [9] FAÍSCA, R., Characterisation of dynamic loads associated with human activities, PhD. Thesis, Universidade Federal do Rio de Janeiro, Brasil, 3 (in Portuguese). [] GRUNDMANN, H., KREUZINGER, H. & SCHNEIDER, M., Schwingungsuntersuchungen für Fußgängerbrücken, Der Bauingenieur 68, 993 (in German). [] BACHMANN, H., PRETLOVE, A. & RAINER, H., Dynamic forces from rhythmical human body motions, in Vibrating roblems in structures-ractical guidelines (aendix G), Birkhäuser Verlag,.85-9, 995. [] SYNPEX Semestrial Reort no., Contributions from RWTH Aachen and SBP, Nov. 5.