1 INTRODUCTION

In recent decades there has been a trend towards improved characteristics of materials used in footbridge construction, enabling to design lighter and slender structures. In turn, these footbridges are thus more sensitive to vibration problems.

On June 10, 2000, a new footbridge over London's Thames river was opened to the public. Designed by a team including renowned sculptor Sir Anthony Caro and Britain's leading architect, Lord Norman Foster, the Millennium Bridge was built with an extremely shallow profile, intended to resemble “a blade of light”. It was constructed during 1998-2000 at a cost of £18.2 million, including a £2.2 million cost overrun. As an eager crowd streamed onto the bridge for the opening celebration, something went wrong. Within minutes, the bridge developed large amplitude side-to-side oscillations, and the crowd simultaneously began to fall into step. Due to this completely unanticipated motion, city authorities were forced to close the bridge just two days after its inauguration. During the following 18 months, Arup, the engineering firm that built the bridge, spent £5 million to develop a system of passive dampers aimed at controlling the unwanted wobble. Their testing and modelling led to a partial understanding of the problem, but left several interesting phenomena – including the apparently spontaneous synchronization of the pedestrians – unexplained.

In the nineties already, several researches (Grundman et al. 199), (Fujino et al. 1993) noticed that pedestrians are forced to adjust their step length and speed to some extent to the motion of other pedestrians if footbridges are exposed to large pedestrian traffic: this mechanism can be defined as synchronization among pedestrians and it is independent of the footbridge dynamic properties. In the case of footbridges with a lateral frequency close to the walking frequency (i.e. 1 Hz), a further synchronization can be achieved between the motion of the footbridge and pedestrians (Fujino et al. 1993). The excessive lateral sway motion caused by crowds walking across footbridges has attracted great public attention in the past few years: the example of the Millenium footbridge is not the only one and it is possible to mention other footbridges which

Dynamic investigations of the Solferino footbridge

C. Cremona Direction for Research and Innovation Ministry for Energy, Ecology, Sustainable Development & Spatial Planning, La Défense, France

ABSTRACT: As footbridges increase in span, flexibility and lightness, they should be designed with greater care regarding vibration phenomena. Dynamic design rules and criteria related to pedestrian comfort become essentials. A description of both experimental and numerical investigations of the dynamic behaviour of the Solferino footbridge induced by pedestrian walking (moving crowd) is presented. This paper presents investigations results obtained in 2003 on the footbridge. The accelerations induced by the loading generated by a large number of pedestrians walking in various configurations (one way crossing, go and back and circular walking) were measured. The aim was to determine the critical number of pedestrians needed to produce significant vibrations and to explore the phenomena of synchronization between the walk of the pedestrians and the horizontal vibrations of the footbridge.

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were prone to these vibrations: T-bridge in Japan, Solferino footbridge, … In this latter case, during its opening day in December 1999, unexpected lateral movements occurred as pedestrian were crossing the footbridge. Consequently, 14 tuned mass dampers (6 TMD for the horizontal movement and 8 vertical for the torsion one) were set at mid span to reduce the excessive vibrations, without compromising the original design.

In 2002, several tests were carried out in order to check the efficiency of the TMDs and to get some information (not operating TMDs) when excessive crowds were crossing the bridge. This analysis, supported by the engineering office (SETRA) of the Ministry of Transport, was carried out by LCPC and CSTB (Cremona et al. 2003a-b). For this purpose, the footbridge was instrumented with 14 accelerometers, 7 additional accelerometers being installed during the crowd tests. The first series of tests were performed by using harmonic forces induced by a mechanical actuator. The TMDs were operating during these tests, the objective being to identify modal properties and especially modal damping in order to assess the serviceability of the footbridge. The second series of experiments used moving crowd tests (from 60 to about 400 participants). In this case, in order to reproduce excessive lateral vibrations, the TMDs were inactive (locked). This paper presents the different experimental results obtained from the two series of tests.

2 THE SOLFERINO FOOTBRIDGE

The Solferino footbridge is a double-deck structure constituted by two double arches (Fig.1). The footbridge was designed by the architect Marc Mimram and built by the Eiffel company from June 1995 to June 1999. The main structure is 140 m long (with a main span of 106 m) and from 11 to 15 m large. The steel weight is approximately 990 t.

Tuned Mass Dampers

a) b) Figure 1: View of the a) Solferino footbridge and b) TMDs

Figure 2: Location of the TMDs

Due to the excessive lateral vibrations, several solutions were analyzed to reduce them. It

was considered impossible to stiffen the structure for aesthetical reasons but also for technical

Main arches

Spring damper

Pendulum damper

Longitudinal axis of the footbridge

Existing structure New added structure

65

ones. Indeed, the use of horizontal stiffeners would induce frequency shifts from 0.8 Hz to 1.0 Hz, and would reduce by 40% the vibration amplitudes, but they would not change the acceleration levels, and consequently, would not improve the pedestrian comfort. An alternative was to increase the modal damping by adding tuned mass dampers (TMD).

a) b) Figure 3: a) Finite element model and b) comparison between damped and undamped systems

1st lateral 0.81 Hz 1st bending 1.12 Hz

Torsional 1.97 Hz Torsional/lateral 2.07 Hz

Figure 4: Example of mode shapes identified by the FE model of the Solferino footbridge

For the footbridge (Fig.1b, Fig.2), this system was considered for the most uncomfortable modes (horizontal close to 1 Hz, vertical close to 2 Hz). For the first case, simple pendulums can be installed at mid-span while in the second case, suspended masses on both sides of the footbridge can be used. Based on numerical simulations and preliminary experimental investigations, it was recommended to use 6 pendulums (2500 Kg) and 8 suspended masses (2x4) for damping the mode at 1.94 Hz (4x2500 Kg masses) and the mode at 2.22 Hz (4x1900 Kg masses). A finite element model (Cremona 2003) was developed to serve as a basis for the experimental investigations (Fig.3a). This model was made with the CESAR-LCPC© software and is composed of 4436 nodes and 7858 elements (5472 “2-nodes” beams and 2386 “4-nodes” plates). This finite element model was also used to assess the efficiency of the TMDs (Fig.3b). Fig.4 presents several undamped mode shapes determined by the dynamic analysis of the FE model.

Damped Undamped Real

Frequency (Hz)

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3 IDENTIFICATION OF THE MODAL PROPERTIES

The instrumentation consists of 21 accelerometers set horizontally and vertically at different locations on the bridge deck, including two accelerometers set on two TMD masses. Solicitations were made with a mechanical exciter (Fig.5), which produces a sinusoidal force at various frequencies and amplitudes. Depending on the orientation of this actuator, vertical or horizontal forces can be applied. The force amplitudes can vary through the range ±400 daN for the vertical force and ±175 daN for horizontal force. A specific assembly of the actuator (sinusoidal translation of the actuator) was used to have a high excitation load at a low frequency. This assembly was used to identify the horizontal swaying mode whose frequency is very low (0.71 Hz) and which is very strongly damped by the horizontal TMD.

a) b)

Figure 5: View of the a) accelerometers and b) of the mechanical excitation device

Two approaches were used to determine the modal properties. The first one is the analysis of the different channels (amplitudes and phases) and the second one is the use of the random decrement (RD) technique. The first technique was introduced to check quickly the resonance peaks during the tests. The RD time domain analysis was employed for identifying the frequencies and mode shapes during harmonic tests and damping ratios from decayed responses after stopping the actuator. Table 1 presents a comparison between calculated and identified frequencies. Two results can be noticed. The first result concerns the non identification of the first bending mode shape. A torsional mode is identified instead with a frequency close to the bending mode. This is essentially due to the fact that it was not possible to physically locate at quarter-span the actuator (a large part of the structure is open because connecting the lower arch to the upper one). Consequently the actuator was not centered on the footbridge and pushes the structure to move with torsional displacement. The second results is related to the 2.01-2.45 Hz frequency range where no frequency is dominant applying an excitation on the structure. This explains that for two series of tests, two separate frequencies were identified although the corresponding mode shapes looked similar. When applying the excitation on the access footbridge (which is connected to the main arch structure), it is nevertheless possible to highlight a frequency close to 2.36 Hz on the main span (small amplitude).

Damping rations have been identified from decayed responses with the help of the logarithmic decrement method (when stopping the actuator motion), or by the analysis of the response transfer functions by the half-power method. Table 2 synthesises the determined damping ratios. Values in Table 2 are mean values; error is related to a 95% confidence interval (assuming a normal distribution). Some mode shapes are given on Fig.6. The actuator tends to change the shape of the mode by increasing the modal amplitude at the actuator location. For some modes excited from the main footbridge, it appears that the access bridge is also excited. This highlights the continuity between the two structures.

The determination of the modal masses for each mode is based on the analysis of a single degree of freedom oscillator under harmonic excitation. In that case the modal mass is given by:

( ) ( )*

2 22 2 2(2 ) 2

φ=

φ π − + ζ

ej ej

rj r j e j j e

Fm

Y f f f f (1)

67

where Fe is the excitation amplitude for the excitation frequency fe. φej and φrj are the modal amplitudes for mode j at the excitation location and ate the response location. Yr is the measured amplitude. fi and �i are respectively the identified natural frequencies and damping ratios. Table 3 provides the modal masses for the first 7 modes. Theoretically, the excitation frequencies are tuned on the structure frequencies by a frequency scanning with the help of the actuator. This scanning is not sufficiently accurate and can lead to frequencies which can be slightly different from the identified frequencies by the random decrement technique for instance (coupled with the Ibrahim Time Domain technique).

Table 1 : Comparison between calculated and identified frequencies

Mode Experimental (Hz)

Numerical (Hz) Characteristics

1 0.71 0.811(0) - 0.70 (1) Symmetrical lateral bending (with torsion) 2 1.09 - Non symmetrical torsion, 1 node, 2 lobes 3 1.12 (2) 1.123(0) Non symmetrical bending, central node, 2 lobes

4 1.55 1.386(0) Non symmetrical torsion, central node (with lateral bending)

5 1.56 1.787(0) Symmetrical bending, 1 lobe

6 1.70 1.527(0) - 1.680(1) Non symmetrical torsion, central node, 2 lobes (with lateral bending)

7 1.95 1.971(0) Non symmetrical torsion, 2 nodes, 3 lobes (with lateral bending)

8 2.01 2.076(0) - 2.31(1) Symmetrical torsion, 2 nodes, 3 lobes (with lateral bending)

9 2.36 (x) Vertical bending of the access bridge, 1 lobe Torsion of the main footbridge

10 2.51(x)-2.60(x) 2.386(0) Vertical bending of the access bridge, 1 lobe

11 - 2.722(0) Non symmetrical lateral, 2 nodes, 3 lobes (with torsion of the access bridge)

12 2.88 2.784(0) Symmetrical bending, 2 nodes, 3 lobes 13 3.02 – 2.92(x) 2.928(0) Torsion of the access bridge, 1 node 14 3.09 3.024(0) Non symmetrical bending, 3 nodes, 4 lobes 15 - 3.204(0) Symmetrical torsion, 3 nodes, 4 lobes

16 3.32 3.42(0) Non symmetrical vertical bending, 1 node, 2 lobes (with movement of the side arches)

17 3.52 3.506(0) Symmetrical bending, 2 nodes, 3 lobes (with lateral)

18 3.64 (x) 3.958(0) Symmetrical bending, 1 lobe (with movement of the lateral arches )

19 3.74 3.385(0) Symmetrical bending, 1 lobe (with movement of the side arches)

20 4.22 - Symmetrical bending, 1 node, 2 lobes (with movement of the side arches)

(0) calculation without TMDs - (1) calculation with TMDs – (2) with crowds (x) identified modes when actuator of the access bridge

Table 2 : Identified frequencies and damping ratios

Frequency (Hz) Damping ratio (%) Frequency (Hz) Damping ratio (%) 0.69 ± 0.02 0.0230 ± 0.005 2.51 – 2.60 ± 0.02 0.0132 ± 0.005 1.09 ± 0.02 0.0090 ± 0.002 2.88 ± 0.02 0.0193 ± 0.005 1.12 ± 0.02 - 2.92 ± 0.02 0.0120 ± 0.005 1.55 ± 0.02 0.0050 ± 0.003 3.08 ± 0.02 0.0127 ± 0.002 1.56 ± 0.02 0.0060 ± 0.002 3.10 ± 0.02 0.0033 ± 0.002

1.69 ± 0.02 0.0250 ± 0.005 3.32 ± 0.02 0.0035 ± 0.002 1.95 ± 0.02 0.0038 ± 0.002 3.52 ± 0.02 0.0031 ± 0.002

2.01 – 2.45 ± 0.02 0.0260 ± 0.004 3.74 ± 0.02 0.0055 ± 0.002

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Table 3 : Estimation of modal masses for the main footbridge

ef (Hz) jf (Hz) rY (mm) jζ (%) eF (N) *jm (T)

0.69 0.698 0.364 2.30 92.5 260 1.09 1.086 1.270 0.90 330 285 1.55 1.53 0.996 0.50 574.3 220 1.56 1.535 1.105 0.60 577.4 163 1.69 1.57 0.250 2.90 599.2 145 1.95 1.912 0.499 0.38 663.11 224 2.01 1.973 0.540 2.60 713.3 216

Although the excitation level and the damping ratio are respectively is very low and very

high for the first lateral mode, this mode is very well identified at 0.69 Hz. Calculations with and without TMDs give respectively 0.7 Hz and 0.81 Hz. The vertical displacement of the lateral pathways represents 30% of the lateral motion at mid-span. Sensors installed on TMDs show that the TMDs masses are not vibrating with the same amplitude (from 1.5 to 2 times the modal amplitude), highlighting that the tuning of the horizontal TMDs is not identical. Close to 1 Hz. two frequencies are identified (1.09 Hz and 1.12 Hz). The first one is a non symmetrical torsional mode. The vertical displacement represents 85% of the lateral displacement with a high participation of the TMDs. This mode is not identified with the numerical simulations. The second one is a non symmetrical bending mode; it is not identified with the actuator but with crowds tests. This mode is highlighted by the FE model. It is assumed that the 1.09 Hz mode is in fact the 1.12 Hz mode, the torsional shape being emphasized by the non centred presence of the actuator on the footbridge. The non symmetrical torsional mode at 1.55 Hz exhibits very large amplitude. Its presence can also be noticed on all the crowd tests. Its damping ratio is low (0.5 %). When the actuator is located at mid-span with a non centred position, a combined torsion/bending mode at 1.56 Hz is identified. The influence of the actuator is noticeable on several other modes : it affects the mode shape and induces a slight frequency shift. At 1.7 Hz. a torsional mode with two nodes and two side lobes is identified. The TMDs motion is large (from 1.6 to 2.3 times the modal amplitude).

0.71 Hz 1.09 Hz

1.55 Hz 1.56 Hz

1.70 Hz 1.95 Hz

Figure 6: Identified mode shapes

69

2.01 Hz 2.36 Hz

2.51 Hz 2.88 Hz

2.92 Hz 3.09 Hz

3.32 Hz 3.52 Hz

Figure 6 (continued): Identified mode shapes The FE model allows estimating these modes (without and with TMDs) between 1.527 and

1.68 Hz. Between 1.8 to 2.6 Hz. no dominant mode is highlighted during the vibration tests. Two different frequencies have been identified (i.e. : 2.02 and 2.08 Hz) but their mode shapes look identical. The vibration tests on the access bridge help to identify a global mode at 2.36 Hz (bending mode for the access bridge and torsion mode for the footbridge). The maximal amplitudes are low (± 0.75 m/s²). The local modes for the access bridge are identified around 2.66. 3.02 and 3.64 Hz. Above 3.0 Hz. a large number of modes are present and several modes between 3.7 and 4.5 Hz exhibit a firefly shape. Damping ratios for these modes are relatively low.

4 CROWD TESTS

These experiments were conducted for different group sizes (with a maximum of 400 persons) for various configurations. The bridge vibrations induced by the pedestrians were measured. An electronic beeper generating audible signals at various frequencies was used in some experiments (rhythmic tests) to facilitate the pedestrians to synchronize their footstep frequency (walk at fixed pacing frequencies). Free tests were also realized, the pedestrians moving at their own pace. All those experiments were performed with inactive horizontal TDMs, for identifying lateral vibration effects. Some tests were carried out with a constant number of participants, others with a gradual increasing number. Fig.7 provides a view of different crowd tests performed on the Solferino footbridge on October 21st 2003.

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Table 4 summaries the different tests realized during this program. Fig.8 presents some time histories of the horizontal accelerations. Among the different crowd

tests, series 1 and 2 are the most interesting ones. The first series highlights the acceleration levels and the pedestrian synchronization. For test 1a, the maximal horizontal acceleration is ± 0,55 m/s² for the lateral mode at 0.7 Hz. This value is far above the usual threshold comfort criterion of 0.1 m/s². This value is reached for a number of critical pedestrians equal to 138.

a)

b)

Figure 7: Examples of crowd tests: a) circular, b) frontal

Table 4: Categories of crowd tests Test Characteristics Number of pedestrian Beeper 1a circular and very slow walk 69, 138, 207 no 1b circular and slow walk 92, 115, 138, 161, 184, 202 no 1c circular and slow walk + beeper 114, 160, 229 0,75 Hz

2a1 simple frontal slow walk 229 no 2a2 simple frontal slow walk 160 no 2b return walk, fast walking 160 no

3a11 frontal walk + beeper 368 1,10 Hz

3a12 frontal walk + beeper 184 1,60 Hz 3b11 crossing walk + beeper 184 + 184 1,10 Hz 3b12 crossing walk + beeper 184 + 184 1,60 Hz 3b13 crossing walk + beeper 184 + 184 2,00 Hz 3b14 crossing walk + beeper

184 + 184 2,40 Hz

Figure 8: Examples of time histories

a) b)

Figure 9: a) Frequency and b) amplitude synchronization

71

a) b) Figure 10: Influence of the walking pace a) slow, b) high

The global analysis of these signals does not allow understanding the synchronization

phenomena in the structure vibration problem. A time-frequency analysis of the lateral signals show that the synchronization is more and more dominant as long as the number of pedestrians increases (Fig.9a). Comparing the vibration levels at each measurement points on the footbridge emphasizes that the lateral mode (0.71 Hz) is dominating (Fig.9b).

The second series of tests (2a, b) highlights the importance of the crowd pace regarding the synchronization phenomenon. For the first two tests (Fig.10a), for low walk speed, synchronization is explicit, whereas for a quicker walk with the same number of people no synchronization occurs (Fig.10c)

In conclusion on the different tests, the horizontal swaying mode is more easily excited when a sufficient number of pedestrians is walking slowly on the bridge (i.e. when their walking frequency is slightly smaller than the structure frequency). The maximal horizontal accelerations obtained during these tests were ± 1 m/s² (frontal walk) where 229 pedestrians were walking from on side to another (2a tests) exciting the swaying mode at 0,71 Hz.

From these diferent tests, it is interesting to identify if the same correlation between deck velocity and human lateral excitation as highlighted for the Millenium footbrigde (Dallard et al. 2002) can be obtained on the Solferino footbridge. Applying the same methodlogy than Dallard et al, the modal (generalized) force for the frequencies at 0,7 Hz and 1,56 Hz, is calculated and compared to the modal (generalized) velocity. Fig.11 provides these relations for the mid-span point (tests 1a).

Forc

e/pe

rson

(N

)

Velocity (m/s)

Forc

e/pe

rson

(N

)

Velocity (m/s) Figure 11: Relation between human excitation and bridge velocity

Fig.11 highlights a strong correlation for the first lateral mode and no correlation for the first

vertical mode. It is almost identical to the Millenium footbridge. Furthermore, the linear fitting provides values comprised between 550 and 650 N/ms-1. This value is very close to the one given by Dallard ( -1300 2 600 N/ms× = ). The latter value is given in terms of physical coordinates although in this paper it is given in generalize (modal) coordinates. To compare the two values, Dallard’s value must be multiplied by modal norm (2 in the case of the Millenium footbridge). The analysis of the modal accelerations show a logarithmic linearity versus time and two degrees of synchronization (Fig.12). Three domains can be distinguished. The first one (I) is characterized by a lack of synchronization. The second (II) is a partial synchronization at the points of large modal amplitudes. The third (III) appears when all the spatial points are synchronized on the mode shape.

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I II

III

Time (s)

Acc

eler

atio

n (m

/s2)

Figure 12: Synhronization domains

5 CONCLUSIONS

The different vibration tests performed in 2002/2003 on the Solferino footbridge has allowed determining its modal properties and the TMDs efficiency. The crowd tests have helped to provide a better understanding of the human-induced synchronization on the solferino footbridge. In particular it was noticed that 140-160 pedestrians can initiate large levels of lateral accelerations making the footbridge uncomfortable. Due to the architectural shape of the footbridge, this synchronization is made in two stages and the crowd walking frequency close to the structure frequency. It is even noted that synchronization occurs if the walking frequency is smaller than the structure frequency. As mentioned in introduction, several tentative have been made to model human-induced vibrations on footbridges. The mechanisms investigated in the literature can be classified into three classes:

• direct resonance activated if pedestrian excitation is in resonance with a mode of vibration,

• dynamic interaction based on suitable models of the interaction between the bridge motion and the pedestrian motion,

• internal resonance due to structural nonlinearities giving rise to internal resonance conditions among different modes of vibration.

Dynamic interaction is certainly the family of models which receive most of the attention, but none of the proposed models can explain the physical phenomenon of synchronization and strongly rely on experimental data and empirical formulas. An efficient and pertinent model for human induced vibration cannot be built without clearly providing an understandable meaning to the synchronization phenomenon. Consequently, any model which does not attempt to model this synchronization will fail to be an appropriate model for analyzing footbridge vibrations under pedestrian loading (Cremona, 2008).

RÉFÉRENCES

Cremona C., Danbon F., Grillaud G., and Habib-Hallak P., 2003 – Dynamic monitoring of the Solférino footbridge – Part A : instrumentation and vibration tests. EN-CAPE 03.028C; 2002\33 018 902

Cremona C., Danbon F., Grillaud G., and Habib-Hallak P., 2003 – Dynamic monitoring of the Solférino footbridge – Part B : crowd tests. EN-CAPE 03.028C; 2002\33 018 902

Cremona C., 2008 – New trends and new models for analyzing dynamic interactions, 17th Congress of IABSE, Chicago, September 17-19 2008, 35-111

Dallard P., Fitzpatrick AJ., Flint A., Le Bourva S., Low A., Ridsdill Smith R., and Willford M., 2002 – The Millenium footbridge in London, Bulletin Ouvrages métalliques, 2, 63-91.