The dynamic loads developed by pedestrians

Transcription

1 FEATURE Dynamic Testing of Civil Engineering Structures Series CABLE TENSIONING CONTROL AND MODAL OF A CIRCULAR CABLE-STAYED FOOTBRIDGE by C. Rebelo, E. Júlio, H. Varum and A. Costa The dynamic loads developed by pedestrians crossing a footbridge are characterized by the superposition of only few harmonics, multiples, and submultiples of the pace rate. 1 Although the simultaneous effect of several pedestrians depends on their synchronization, in normal circumstances it tends to be a problem of resonance in a vibration mode with significant vertical or horizontal displacements. When the vibration of the deck is perceptible for the pedestrians, synchronization of the human movements and the deck vibration is likely to occur. 2,3 Recently a number of footbridges were built in Portugal within the scope of the urban renovation plans of several cities. This article deals with the modal identification and vibration measurements and finite element (FE) modelling of one of these structures located in Aveiro, Portugal. The circular plan view of the bridge allows the link among the three points located on the banks of the T-shaped junction of two canals (Figs. 1 and 2). The vibration measurements are part of the on-site structural characterization of the bridge that also included load tests, terrestrial photogrammetric survey, and long-term strain measurements using Brag sensors located on the cables and on the mast. BRIDGE DESCRIPTION The footbridge is composed of a circular deck supported at three points and suspended from eight cables connecting it to the extremity of a steel mast anchored to a concrete bulk situated on the riverside (Figs. 2 and 3). The circular deck with 13-m radius is made of a 2-m wide wooden floor supported by a steel structure composed of 55 modules. Each module is a 1.5-m long steel girder made of HEB200 longitudinal profiles and HEB100 radial and diagonal profiles. The whole deck is very stiff in its own plane Editor s Note: This article is part of the ongoing Feature Series on Dynamic Testing of Civil Engineering Structures. This series covers a wide range of technologies appropriate to civil engineering structures from both practical/technical and analytical perspectives. Series editor: Paul Reynolds, The University of Sheffield. C. Rebelo (SEM member; crebelo@dec.uc.pt) and E. Júlio are assistant professors with ISISE, and assistant professors at the Department of Civil Engineering, University of Coimbra, Portugal, H. Varum is an assitant professor and A. Costa a full professor of the Department of Civil Engineering, University of Aveiro, Aveiro, Portugal and horizontal movements are hindered at the supports, so that no modification of the circumferential form is expected during serviceability loading from pedestrians. The bow-shaped mast is a tube with a quadrangular tapered steel-welded cross-section. The section width varies from 0.60 m, near the fixed supports, to 0.43 m, at the top. The section height is constant and equal to 0.43 m. The mast is stabilized by two steel strips anchored on a bulk at the riverside (Fig. 3). The stay-cables are connected to the top of the mast (Fig. 4). Each steel strip has a mm 2 rectangular cross-section, and the cables are made of 36-mm diameter steel bars with lengths given in Table 1 and Fig. 5a. TESTS AND MEASUREMENTS The measurement system used consisted of eight accelerometers with the characteristics given in Table 2, one PC with LAN interface, PULSE software, and IDA-based data acquisition front-end hardware (PULSE Type 3560/D from Brüel & Kjær, Nærum, Denmark). The vibration measurements on the bridge were planned aiming: (1) to ensure the correct installation of axial forces in the cables; (2) to characterize the dynamic behavior of the bridge; and (3) to obtain the response acceleration when groups of synchronized pedestrians cross the bridge. Before performing the vibration measurements for the modal identification, it was necessary to control the introduction of the correct forces in the stay-cables to ensure the structural behavior of the bridge assumed in design. Therefore, dynamic measurements were performed simultaneously in all cables during the tensioning operation to obtain in real time the natural frequency of the cables which is directly related to the tension force installed. The next step and main objective was the characterization of the dynamic parameters through output-only modal identification techniques to calibrate an FE model. A total of 14 measurement points were established along the bridge deck. The data acquisition was performed using five roving and three reference accelerometers, including one reference vertical accelerometer at the top of the mast. The modal extraction was conducted using ARTeMIS software (Structural Vibration Solutions A/S, Aalborg, Denmark). 4 After this stage, groups of pedestrians crossed the bridge at different speeds to obtain acceleration response histories doi: /j x 62 EXPERIMENTAL TECHNIQUES July/August , Society for Experimental Mechanics

2 Fig. 4: Top of the mast Fig. 1: Localization of the bridge the natural frequencies of cables. The force could be easily changed by means of a screw system installed in the cables. The natural frequency (f n ) corresponding to the n th mode of vibration of a tensioned cable can be obtained in Hz from a simple equation (Eq. 1) relating it to: the axial force (T), the mode of vibration (n), the mass per unit length (m), and the cable length (L). Fig. 2: Circular footbridge f 2 n = Tn2 4 ml 2 (1) Actually, the precision of this expression, based on the theory of vibrating strings, depends on the mode of vibration that is being evaluated. 5 For the first mode, the frequency is obtained with a precision usually lower than for the higher modes. The parameter λ 2,givenby: λ 2 = (9.8 ml) 2 EA T 3 (2) Fig. 3: Side view of mast, stay-strips and stay-cables and estimates for maximum response accelerations that can occur during bridge lifetime. Stay-cable Forces The planned structural behavior of the bridge deck depends on the correct tension forces installed in the cables. To ensure that these forces correspond to those predicted by the designer, a simple procedure to control the cable forces can be carried out if the natural frequencies of the cables are monitored during prestressing. This was done by measuring the accelerations simultaneously on all cables using eight measurement channels and a frequency analyzer to estimate is a measure of the geometric characteristics and of the deformability of this type of cable. Values of λ 2 < 1 usually indicate that the error in Eq. 1 is very low. The higher the stress in the cable, the higher the accuracy of this expression. For the cables used in the bridge (Table 1 and Fig. 5a), the values vary from approximately 0.5, for cables 1 and 8, to 1.4, for cables 2 and 7. All the other cables present values < 1. In the present case, the use of the first natural frequency of the cables in Eq. 1 gives results with acceptable precision, and therefore it was used to estimate the corresponding tension force in the cables. The reason why the first natural frequency was adopted instead of a higher frequency is that some construction works were still being performed at the same time, hindering accurate estimates of higher natural frequencies of the cables. After several iterations concerning the force introduced in each cable, the results given in Table 1 were obtained. A sufficiently good approximation between measured values and those predicted by the designer was observed. July/August 2010 EXPERIMENTAL TECHNIQUES 63

3 Table 1 Stay-cables length and respective axial forces for dead loads CABLE NUMBER Cable length (mm) st frequency (Hz) Estimated force (kn) Design force (kn) Fig. 5: Cable numbers (a) and measurement points on the bridge deck (b) Table 2 Characteristics of the accelerometers SENSITIVITY FREQUENCY QUANTITY TYPE (MV/G) RANGE (HZ) 4 B&K PCB 393B Modal Identification The modal identification was based on the techniques of modal extraction in the frequency and time domain implemented in ARTeMIS software, 4 namely the enhanced frequency domain decomposition and the stochastic subspace iteration methods. These techniques, implemented in the referred software, allow the estimation of modal damping, natural frequencies, and mode shapes of the deck 6,7 using measured acceleration responses of the bridge at several locations. Because these techniques assume broad band white noise excitation, the measurements need not be performed simultaneously, given that some of the sensors remain at the same position for all measurement setups. The vertical acceleration induced by the ambient excitation mainly because of wind was measured on the bridge deck and on the top of the mast. To ensure an accurate definition of the mode shapes, 14 measurement points were considered (Fig. 5b), three of them with reference accelerometers (points 1, 6, and 7) and the remaining 11 with roving accelerometers. 64 EXPERIMENTAL TECHNIQUES July/August 2010

4 Steel strips Bridge deck Acceleration[Db] Stay cables Frequency [Hz] Fig. 6: Average of the normalized singular values of spectral density matrices of all test setups and labelled first natural frequencies of structural elements Based on previous experience, setups with duration of about 1000 times the expected highest natural period allow good quality modal estimations. In this case, the highest expected natural period should be about 0.6 s corresponding to FE modelling-based frequency estimation of 1.7 Hz. Therefore, a total of three setups with 10-min duration each were used to obtain the ambient vibration response at each measurement point necessary to identify the modal parameters. The results of the experimental modal analysis are summarized in Fig. 6 and in Table 3. The natural frequencies of the steel strips behind the mast appear very clearly in the range Hz. Because Eq. 1 is a good approximation also for the natural frequencies of these strips, it was possible to calculate the corresponding installed tension forces using these frequencies and to verify that these correspond to the values predicted by the designer. It should be noted that the values are not exactly the same for both strips because the structure is not perfectly symmetrical in plane view. The peaks corresponding to the first natural frequencies of the cables can be seen in Fig. 6 in the frequency range Fig. 7: Mode shapes of the deck experimental versus numerical results from 2.2 to 2.8 Hz. The peaks in the higher frequencies correspond to the natural frequencies of the deck and to the second natural frequencies of the strips and cables. The four frequency peaks at 3.17, 3.25, 3.96, and 4.05 Hz correspond to approximately symmetrical or antisymmetrical mode shapes of the deck represented in the left-hand side of Fig. 7. Based on the initial design model, an FE model was developed with the software Robot Millennium 8 (Robobat, Paris, France) using steel be m elements. In order to match the natural frequencies of the deck given in Table 3, Table 3 Experimental natural frequencies and damping, numerical frequencies, and mode shapes FREQUENCY (Hz) DAMPING (%) MEAN SD MEAN SD FREQUENCY IN FE MODEL (HZ) MODE SHAPE Steel strip supporting the mast Steel strip supporting the mast Mode 1 of the deck Mode 2 of the deck Second mode of strip Mode 3 of the deck Mode 4 of the deck July/August 2010 EXPERIMENTAL TECHNIQUES 65

5 the support stiffness on the river banks were considered variable parameters. The respective mode shapes were shown in the right-hand side of Fig. 7. Although all these modes involve flexural and torsional movements of the deck, the first two modes are mainly flexural modes and the others present higher contribution of torsion. It should be noted that because of the measurement setup used, the graphical representations of the experimental mode shapes in Fig. 7 include both bending and torsion movements of the deck. Human-induced Dynamic Response The bridge is located in a newly rehabilitated part, near the city center, surrounded by large green spaces and walkways, where it is expected to have groups of people walking or jogging during holidays and weekends. To simulate the expected loading situations, a large group of university students were invited to collaborate in the tests to be performed on the bridge involving pedestrian loading. The vertical accelerations at points 1 8 (Fig. 5) were recorded when the group of 65 people (about 46 kn) circulated on the bridge in synchronized walking and running. Taking into account that no lock-in effect because of horizontal movements of the deck were expected and that the major concern was the vertical vibration of the deck and steel strips, the number of people on the deck was maintained constant during the forced vibration tests. Actually, control measurements with smaller groups were also performed and accelerations were always lower. The pace rate was initially chosen to be simultaneously in the range of the normal/slow walking and of the steel strips first natural frequency, to assess the sensitivity of the bridge and particularly the steel strips to the vibration induced by day-to-day normal usage of the bridge. Therefore, the value of 1.8 Hz was chosen. In order to excite the natural frequencies involving vertical movement of the deck, further experiments with people running at pace rates of 3.0 and 3.5 Hz were carried out. Although these frequencies did not match the natural frequencies of the deck, which were identified a posteriori as being exactly 3.17 and 3.25 Hz, they could provide representative load spectra in the range of interest. It should be highlighted here that, despite the external sound beat used to synchronize runners, such pace rates corresponding to normal to rapid running were not expected to provide single sharp spectral load peaks. Instead, the input energy was spread around those beat frequencies as expected (Fig. 8). Moreover, after the initial seconds, runners distributed more or less along the deck because the bridge is circular and people did not leave the bridge during the test. Therefore, unavoidable phase differences among runners did not allow the resonant in-phase excitation of the first and second modes. Nevertheless, these were considered representative extreme dynamic load situations for the excitation of the bridge deck. The maximum values for acceleration and displacement obtained from the records after band-pass filtering between 0.5 and 25 Hz are summarized in Table 4. The time histories had durations varying between 8 min, for the lower pace rate, and 1 min for the highest one. The Eurocode EN gives the indicative maximum acceleration value of 0.7 m/s 2 for vertical movements of the deck during normal use. Other guides and codes 10,11 give values of up to 1 m/s 2. Considering that the first loading case in Table 4 fulfils the requirement of normal usage, the measured maximum values of acceleration are borderline. However, if higher pace rates are considered, for instance, 3.0 Hz corresponding to normal running or 3.5 Hz corresponding to rapid running, it can be concluded that the structure is prone to suffer much higher accelerations (Table 4), which are beyond minimum acceptable comfort levels proposed in recent design guides. 11 Although not measured, the vibrations in the stay-cables also increase significantly for those pace rates. As a consequence, fatigue problems may arise in the connections between cables or steel strips and the mast or deck. This becomes clear when analyzing Fig. 6, where several peaks, corresponding to the natural frequencies of stay-cables, match the pace rates that are expected to occur during the service life of such a structure, that is, between 1.8 and 2.2 Hz. Furthermore, it is not to exclude that a situation of parametric excitation of the steel bands may occur for pace rates of about 3.4 Hz. DISCUSSION AND CONCLUSIONS Vibration measurements and modal identification of footbridges as part of commissioning tests are very important tools to define and prepare necessary surveillance and eventual maintenance and repair works during their service life. The structure analyzed in this study presents an unusual form: a deck with circular plan view supported by staycables and a mast anchored by two steel strips. The complex dynamic response of such a bridge to pedestrian dynamic loadings is difficult to foresee during design, mainly because of the interaction between the stay-cables and the deck itself. The measurements carried out on this bridge were important in several ways. First, they allowed the introduction of the right forces in the cables, so that the structural response could be the one estimated during design. Then, the modal identification of the bridge allowed an accurate interpretation of the dynamic response to the pedestrian loads. This also enabled to predict whether problems of excessive vibrations can occur or not. Finally, indicative maximum accelerations could be obtained for loading situations which the authors think can resemble, on one side, common dayto-day situations during the service life and, on the other side, extreme but still probable dynamic excitation induced by groups of people running. Although the results concerning probable maximum response accelerations of the deck for normal walking are borderline and for rapid walking and running are somewhat excessive, 66 EXPERIMENTAL TECHNIQUES July/August 2010

6 2.0 Acceleration [m/s2] a) Frequency [Hz] Acceleration [m/s2] b) Time [sec] Fig. 8: Acceleration Fourier spectrum (a) and acceleration time history (b) at location 2 for running Table 4 Maximum accelerations (Accel.) and displacements (Disp.) obtained from the time histories after band-pass filtering between 0.5 and 25 Hz SENSOR Walking 1.8 Hz; duration: 8 min Accel. (m/s 2 ) Disp. (mm) Running 3.0 Hz; duration: 3 min Accel. (m/s 2 ) Disp. (mm) Running 3.5 Hz; duration: 1 min Accel. (m/s 2 ) Disp. (mm) Maximum values are given in bold they should not be interpreted as being impeditive of a normal usage of the bridge. However, the bridge owner should be aware that in some situations there can be a live response of the structure, which may frighten the users. Furthermore, attention must be paid to fatigue problems that can occur near the joints. Periodic inspections of strips and cables connections are strongly recommended. ACKNOWLEDGEMENTS The authors would like to thank the design team, Eng. Domingos Moreira and Arq. Luís Viegas, for the structural design information. References 1. Bachmann, H., and Amman, W., Vibrations in Structures Induced by Man and Machines, IABSE, Zürich, Switzerland (1987). 2. Brincker, R., Andersen, P., and Jacobsen, N.-J., Automated Frequency Domain Decomposition for Operational Modal Analysis, Proceedings of IMAC XXV, Orlando, FL; February 19 22, (2007). 3. Zivanovic, S., Pavic, A., and Reynolds, P., Vibration Serviceability of Footbridges Under Human-induced Excitation: a Literature Review, Journal of Sound and Vibration 279:1 74 (2005). 4. SVS, ARTeMIS Extractor Pro, Release 4.1, Structural Vibration Solutions, Aalborg, Denmark (2007). 5. Irvine, H.M., Cables Structures, MIT Press, Cambridge, MA (1981). July/August 2010 EXPERIMENTAL TECHNIQUES 67

7 6. Dallard, P., Fitzpatrick, A.J., Flint, A., et al, London Millennium Bridge: pedestrian-induced Lateral Vibration, Journal of Bridge Engineering ASCE 6(6): (2001). 7. Brinker, R., Zhang, L., and Andersen, P., Modal Identification from Ambient Response Using Frequency Domain Decomposition, Proceedings of IMAC XVIII, San Antonio, TX; February 7 10, (2000). 8. ROBOT Millennium User s Manual, version 20.0, Robobat, Paris, France. 9. CEN, EN1990-Annex2 Eurocode Basis of Design (2001). 10. International Standard Association, ISO 10137: Bases for Design of Structures Serviceability of Buildings and Walkways Against Vibrations, 2nd Edition, Geneva, Switzerland (2007). 11. SÉTRA / AFCG, Footbridges Assessment of Vibrational Behaviour of Footbridges Under Pedestrian Loading Practical Guidelines, Association Française de Génie Civil, Paris, France (2006). 68 EXPERIMENTAL TECHNIQUES July/August 2010