Dynamic Performance of the Guarda Footbridge Carlos Rebelo PhD, Assistant Professor, crebelo@dec.uc.pt Hugo Pimenta MSc student, hugo.pimenta@netvisao.pt Luís Simões da Silva Full Professor, lssilva@dec.uc.pt ISISE, Department of Civil Engineering, University of Coimbra, 3030-788 Coimbra, Portugal ABSTRACT In this paper a summary of a currently used methodology to prevent vibrations problems in footbridges is presented and applied to a case study, a steel cable stayed footbridge built over a road junction in Guarda, Portugal. The complexity of the bridge behaviour and the high uncertainty related to the damping and the natural frequencies estimation support the decision to additionally carry out a modal identification. Accelerations measured during ambient vibration and during free vibration induced by initial imposed displacements allowed the dynamic characterization of the structure in terms of mode shapes, natural frequencies and modal damping in the frequency range of interest, that is, up to about 5Hz. A FE model of the bridge was updated according to the measurement results and dynamic loads were applied in order to compute the response accelerations. Introduction The structural solutions adopted nowadays for the footbridges are characterized by increasing slightness and flexibility and by low structural damping. The most important consequences are the higher sensitivity to the dynamic loads developed by the pedestrian traffic and the probable excidence of the vibration levels that are considered comfortable for the users. The occurrence of extreme vibrations in footbridges can be avoided either by following a proper structural conception concerning the natural frequencies or by previewing special devices that allow the reduction of the vibrations to acceptable values. The use of passive Tuned Mass Dampers (TMD s) is a technique that has being successfully used in footbridges since several years. Such devices are used both to solve unforeseen excessive vibrations and as a tool to allow more audacious solutions when included in the structural conception and design of the bridges. These devices are simple one-degree-of-freedom oscillators constituted by one mass, which is usually about 3% to 5% of the modal mass of the structure for which resonance is expected, and by springs and dampers designed according to clearly established criteria [1,4,7]. Design methodology for footbridges Recently the French authority for bridges, SÉTRA Service d Études techniques des routes et autoroutes, presented a methodology [7] that can be used by footbridge designers and owners in order to define whether a dynamic analysis is needed and the respective dynamic loads to be considered during the design. This methodology is summarized in the flowchart of Fig. 1 and relies on the initial classification of the footbridge depending on the expected pedestrian traffic and on the comfort level that must be assured for that traffic.
Designer / Owner Bridge class traffic intensity Comfort level Class I to III Class IV Natural frequencies Ressonance hazard No dynamic analysis needed Maximum allowed acceleration to 3 Level 4 Dynamic load cases Maximum response acceleration (a máx ) No dynamic analysis needed Results and conclusions Preventive measures (if necessary) Fig.1: Flowchart for the methodology SÉTRA (adapted from [7]) Footbridge classification The classification of the footbridges is made according to the type and intensity of the pedestrian traffic and, therefore, strongly conditioned by the geographic location (see Table 1). Classification as Class IV implies that the footbridge will not be checked concerning its dynamic behaviour and, therefore, should not be used for slender structures. In this case Class III should be assumed in order to have, at least, results from the modal analysis to support the decision concerning the dynamic response analysis. Class I Class II Class III Class IV Table 1: Footbridge classes Footbridges linking urban areas where very high density traffic is expected or located nearby infrastructures where high concentration of people is possible (e.g. transportation modal interfaces or sport arenas) Footbridges linking urban areas where high density traffic is expected to fulfil the entire bridge deck even if during short periods of time Footbridges for normal use which may occasionally be crossed by large groups of people, although there is a low probability that those groups are able to fulfill the entire deck. Footbridges outside urban areas usually used by single pedestrians or, seldom, by small groups of people (e.g. pedestrian bridges over railways or highways) Comfort levels Comfort levels are established according to acceptable or not acceptable horizontal or vertical acceleration values. Four comfort levels are defined in Tables 2 and 3 for each direction. Although these values are always
subjective, they should be used as reference values when dynamic calculations have to be performed during design. The lower level,, corresponds to the maximum comfort, that is, vibrations are almost imperceptible for the pedestrians. The upper level, Level 4, corresponds to unacceptable vibration amplitudes. To avoid the lock-in effect for horizontal vibrations, that is, the synchronization of groups of pedestrians with the lateral movement of the bridge, the horizontal acceleration must be limited to 0,10m/s 2 Table 2: Comfort levels for vertical accelerations Vertical acceleration (m/s 2 ) 0,0 0,5 1,0 1,5 2,0 2,5 higher Level 2 Level 4 Table 3: Comfort levels for horizontal accelerations Vertical acceleration (m/s 2 ) 0,0 0,1 0.15 0.30 0.8 higher Level 2 Level 4 Resonance hazard For bridges of classes 1 to 3 a modal analysis must be performed considering two situations for the masses: i) only the mass of the bridge is accounted for; ii) the mass of one pedestrian per square meter (70kg/m 2 ) is added to the mass of the bridge. The values of the computed natural frequencies are used to define resonance hazard level according to tables 4 and 5, depending on the direction of the main displacements in the modal shape, respectively, vertical or transversal to the bridge axis. Hazard corresponds to the maximum probability of occurrence of resonance while Hazard Level 4 corresponds to the lowest probability and allows the absence of dynamic computations. Table 4: Resonance hazard levels for vertical vibrations Frequency (Hz) 0,0 1,0 1,7 2,1 2,6 5,0.. Maximum Level 2 Medium Medium Low Level 4 Negligible Negligible Table 5: Ressonance hazard levels for transversal vibrations Frequency (Hz) 0,0 0,3 0,5 1,1 1,3 2,5.. Maximum Level 2 Medium Medium Low Level 4 Negligible Negligible Load cases for dynamic analysis When a dynamic calculation is necessary, the load cases are defined according to the bridge class and the resonance hazard class as shown in Table 6. Three load cases are defined corresponding to low density traffic (Case 1), high density traffic (Case 2) and second harmonic effect (Case 3). The generic formulation of the load function is F () t = F cos( 2πf i t) Ν ψ p, i eq (1) and must be established separately for each direction, vertical and transversal to the bridge axis. The relevant load harmonic amplitude, F P,I, the equivalent number of pedestrians, N eq, and the resonance probability ψ, can be obtained from [7].
Table 6: Identification of the load cases for dynamic analysis Resonance hazard level Bridge class 1 2 3 I Case 2 Case 2 Case 3 II Case 1 Case 1 Case 3 III Case 1 -------- -------- The Guarda Footbridge The steel footbridge, shown in Fig. 2, has a total length of 122.22m, composed by a central suspended span of 90.00m length and two additional spans at both sides with lengths of 8.125m and 5.850m. The deck is three meters wide and is made of precasted concrete panels supported by a steel girder. The deck is suspended from two steel arches in its central part. The arches are inclined towards the interior of the bridge, are connected by transversal bars and are linked to the deck at the intersection level. The arches supports allow the rotation in the plane of the arch. The deck is simply supported at the ends and the continuity on both extremities of the walkway is assured by simply supported spans structurally independent from the intermediate suspended part of the deck. Stainless steel bars with 30 mm diameter suspend the deck at points spaced of about 8.20 meters and their tension can be controlled by varying their length up to ±50mm. Figure 2: Guarda footbridge general view Modal identification by ambient vibration The ambient vibration response of the bridge was measured on thirty points in three directions. The Enhanced Frequency Domain Decomposition method [2, 8] was used to extract the modal information, which is summarized in Table 7. Mode Table 7 Modal results from the ambient vibration tests Frequency [Hz] Damping [% of critical] Mode type Mean Std Deviation Mean Std Deviation Mode 1 0.64 0.004 1.27 0.717 1st mode horizontal Mode 2 1.07 0.005 0.89 0.335 2nd mode horizontal Mode 3 1.48 0.014 1.13 0.474 3rd mode horizontal Mode 4 2.19 0.009 0.34 0.148 1st mode vertical Mode 5 3.65 0.014 0.18 0.064 2nd mode vertical Mode 6 3.69 0.034 0.29 0.190 3rd mode vertical Modal identification by forced vibration The free vibration response of the bridge was measured when initial displacements were imposed in the horizontal and vertical directions in the centre and quarter span of the bridge. The modal information extracted from these measurements is summarized in Table 8.
RefS Reference sensor FreeS Free sensor Mode Figure 3: Accelerometers location for ambient vibration measurements Table 8: Modal results from free vibration with initial displacements Frequency [Hz] Damping [% of critical] Mode type Mean Std Deviation Mean Std Deviation Mode 1 0.63 0.005 2.21 0.268 1st mode horizontal Mode 2 1.05 0.016 1.74 0.847 2nd mode horizontal Mode 3 1.46 0.021 1.43 0.452 3rd mode horizontal Mode 4 2.15 0.025 0.79 0.522 1st mode vertical Mode 5 3.50 0.112 0.36 0.156 2nd mode vertical Mode 6 3.57 0.049 0.30 0.099 3rd mode vertical Finite Element analysis A FE model (Fig. 4) was developed using the software SOFISTIK [8] in order to fit the modal information obtained from the measurements. Beam and shell elements were used to model, respectively, the steel structure and the concrete panels. Table 9 shows the natural frequencies and modal masses concerning the mode shapes with transversal horizontal displacements. The accuracy of the model was considered to be sufficient to reproduce the time response acceleration for prescribed dynamic loading. Figure 4 Finite Element model of the footbridge Table 9- Frequencies and modal masses for the transversal modes Mode Frequency [Hz] Modal Mass [kg] 1 transversal 0.65 77168 2 transversal 1.05 100498 3 transversal 1.39 83574
According to [4] and using the measured damping in the first and second modes, the lock-in effect is expected to occur for, respectively, 94 and 148 pedestrians walking synchronous with the first and second natural frequencies. Following the methodology described above (Table 1) this footbridge is classified as Class II. Concerning the resonance hazard, modes 1 and 2 are classified as, mode 4 as Level 2 and the other modes as. Considering only modes 1 and 2 for the dynamic calculations the density of 0,8 pedestrians/m 2 shall be considered [4] corresponding to 184 pedestrians simultaneously on the deck. Using the measured damping of 2.21% for the first mode and 1.74% for the second mode and the load time functions F 1 (t)= 3,708.cos(2π.0,65.t) [N/m 2 ] (2) F 2 (t)= 3,290.cos(2π.1,05.t) [N/m 2 ] (3) maximum response accelerations of 1,10m/s 2 and 0,143 m/s 2 were obtained for the first and second mode respectively. According to Table 3 these values are classified, respectively, as unacceptable and acceptable with possible lock-in effect. As a consequence, special devices, like TMD s, must be foreseen in order to control the vibrations. Conclusions The recent developed methodologies, like the one proposed by SETRA are valuable information for the designers, since a global approach is possible for the evaluation of the risk of excessive vibrations taking into account several indicators concerning the geographic situation of the bridge and the probability of resonance induced by groups of pedestrians. However, the use of this type of methodologies needs to be applied in as many as possible case studies in order to validate the assumptions. Although the Guarda footbridge is still under evaluation in respect to the measurements of the dynamic response due to the passage of groups of pedestrians, it is possible to classify it and estimate its response concerning the probable load scenarios defined according to the SETRA methodology. The numerical results obtained with a FE model fitted with experimental modal information from measurements point towards the probable need for the inclusion of TMD s to control the transversal horizontal vibrations. Bibliography [1] Bachmann H, Amman W Vibrations in Structures Induced by Man and Machines, IABSE, 1987 [2] Brinker R. Zhang L.,and Andersen P, 2000. Modal identification from ambient response using frequency domain decomposition. Proceedings of IMAC 18, the international Modal Analysis Conference, San Antonio, TX,USA [3] CEB, Vibration Problems in Structures Practical Guidelines Bulletin d Information nº209, 1991 [4] CEB-FIB, Guidelines for the design of footbridges, Bulletin 32, fib, Lausanne, Switzerland, 2005 [5] Pimenta H, Rebelo C, Ensaios de Medição de Vibrações em Ponte Pedonal, CMM, Revista Metálica Nº14, 2007 [6] Ramos L, Caetano E, Cunha A, Análise de Modelos de Carga de Pontes Pedonais. Aplicação à Ponte Pedonal da Guarda, CMNE/CILAMCE, 2007 [7] SÉTRA/AFCG Footbridges Assessment of vibrational behaviour of footbridges under pedestrian loading, 2006 [8] SOFISTIK AG Sofistik Structural Desktop, Munich, Germany, 2004-2007 [9] SVS ARTeMIS Extractor pro, release 4.1, Structural Vibration Solutions, Aalborg, Denmark, 1999-2007