4 th International Conference THE NEW FOOTBRIDGE IN CITY OF CRACOW DYNAMIC DESIGN AND FINAL VERIFICATION IN TEST LOADING Krzysztof ŻÓŁTOWSKI Professor University of Technology Gdansk, PL k.zoltowski@bridges.pl Andrzej KOZAKIEWICZ MSC Engineer University of Technology Gdansk, PL a.kozakiewicz@bridges.pl Summary The paper presents design evolution of the footbridge, from architectural form through basic design to finally built structure. The footbridge over the Vistula River just built in Cracow, was selected by the City Authorities as a result of international competition. The winning project was destined to execution. During the early constructing stage many changes were implemented to the basic design. Predictable dynamic behaviour of the superstructure was one of the major targets in this stage. Assumptions, structural changes and numerical simulations performed by the designing team are presented here. Finally, a dynamic test was carried out to verify the assumptions. Theoretical base, methodology and final conclusions are included. Keywords: footbridge; dynamic; structural concepts. 1. Introduction The just opened new footbridge over the Vistula River in Cracow is a result of evolution from aesthetic design to the final engineer structure ready to carry pedestrians. The history of the designing process and detail explanation of the structure is presented by the authors elsewhere in conference proceedings. The following paper presents general principia of structural changes implemented after the basic design, dynamic analysis and related to dynamic design effects. 2. Design 2.1 Evolution of Structural Concept The footbridge was designed as an arch span (span L=- 143m, arch camber F= 15,34m fig. 1), supported on special abutment prepared to carry full horizontal load. The horizontal structural grid consisted of longitude concrete carriageways and cross pipe beams was hung to the arch by vertical (in elevation view) steel rods. The superstructure was placed on abutment made from plies (partly highly inclined). The basic reason to modify the design, came from the contractor who found enormous problems with execution of the abutment structure. In effect the contractor organised a specified additional consulting team, ready to study the problem: Promost Consulting Rzeszów conception works, detailed design of alternative solution for the abutment and the superstructure. KBP Żółtowski conception of alternative superstructure, dynamic analysis and verification of detailed design. ZB-P Mosty Wrocław consulting. In the early stage of analytic work several problems were indentified - with abutment, dynamics and technology of the erection. Finally, the following issues were identified: Problems in the abutment structure as a consequence of unfavourable static system and structural solutions. 1065
Footbridge Dynamics Unfavourable horizontal alignment of carriageway route with the widest part in the middle of the span. (pendulum effect) Weakness of the structure under asymmetric loads. Huge problem with realization of concrete decks. 143 m 3 m 3.6 m 3.6 m 3 m 7.5 m 7.5 m 3 m 3 m Fig.1 Pedestrian Bridge over Vistula River In Cracow primary design After theoretical evaluation of the bridge several basic changes were implemented (fig.2, fig.3): Tie beam structure which takes horizontal component of dead load from the abutment. Reduction of dead weight of span and reduction of temperature load by the change of the deck structure to steel solution. Implementation of horizontal bracing system to lift up stiffness under horizontal and unsymmetrical loads. Implementation of network hangers (spiral cable) to lift up stiffness under vertical unsymmetrical loads. KAZIMIERZ PODGÓRZE 143 m Fig.2 Pedestrian Bridge over Vistula River in Cracow final design To finalize the design, several static and structural works were done. Detailed explanation is published by the authors elsewhere in proceedings. 2.2 Dynamic Aspects of Final Solution of Superstructure The implemented structural solutions bring a big effect in dynamic characteristic of the superstructure. Thanks to the last year experience we can recognize the problem of dynamic excitation and our aim is to avoid any spectacular happening with excessive vibration of a pedestrian bridge. [1]. 1066
4 th International Conference Actually, the pedestrian bridge is the most popular field for spontaneous architectural experiments where the state of art of the bridge design is ignored. The pedestrian bridge considered in the paper belongs to this party. The structural concept based on one central arch and two curved hanging decks with the widest part in the middle of the span is coupled with unfavourable dynamic characteristic. Such a problem requires advanced numeric simulation to predict the response and safety in nature [2], especially that Cracow footbridge is relatively big and placed in spectacular environment. The final design changes radically the conditions for abutment and dynamic characteristic of the superstructure. The horizontal bracing and network system for hangers increase stiffness. Steel decks dramatically reduce the mass of the span. In effect horizontal loads in abutment are reduced and self-frequencies of the span are lifted up. The change of the first horizontal eigen frequency from 1.18 Hz to 2.15 Hz was very important. A generally not favourable reduction of body mass in this special case, in the view with erection problems and dynamic properties of the superstructure is positive. The plot in Fig. 4 presents the first five eigenforms and eigen frequencies of the alternative structure. Fig. 3 Model od span In FEM SOFiSTiK First eigenform f 1 =0,542 Hz Second eigenform f 2 =1,45Hz Third eigenform f 3 =1,805 Hz Fourth eigenform f 4 =2,134 Hz Fifth eigenform f 5 =2,151 Hz Fig. 4 Eigenforms and eigen frequencies.. Form 1 and 5 horizontal. Form2 torsion. Form 3 and 4 vertical. 1067
Footbridge Dynamics In spite of the positive change in basic dynamic properties, the superstructure was still sensitive to dynamic excitation. Therefore further dynamic analysis of the structure was required. 2.3 Numeric Analysis of Dynamic Response Under Pedestrian Action Several numeric simulations were preformed in two groups: Continuous flow of pedestrians, Purposeful action by synchronised crouching Distribution of loads was done on the base of eigen forms. 2.3.1 Response of span under constant flow of pedestrians - numeric simulation Assumptions: Pedestrians always walk with the same pace rate as self frequency of the span. (eigen frequency no. 3,4,5). Load value was evaluated on the crowd density and depended on pace rate [2]. The average body weight was taken as Bw=0,75 kn. Equivalent dynamic load was defined as uniformly distributed [2] Dumping in FEM model was defined by mass and stiffness dumping related to LD= ~5%. Table 1 Max. crowed density if relation to pace rate density [man/m 2 ] Pace rate [Hz] 1,24 1,805 1,08 2,134 1,07 2,151 Numeric processes were done with SOFiSTiK software using time step Newmark procedure. The dynamic load was defined to act in order with deformation form (eigen form). az [m/sec2] 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 Node 210 0.15 0.10 0.05 0.00-0.05 0.0 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0 39.0 40.0 sec [sec] -0.10-0.15-0.20-0.25-0.30-0.35-0.40-0.45-0.50-0.55-0.60-0.65-0.70-0.75 Fig.5 History of max. vertical acceleration in the deck, amax=0,75m/s 2 free walk with pace rate 1.805 Hz. Number of M synchronized walkers from all N pedestrians is defined by Matsumoto M N 1068
4 th International Conference 2.3.2 Numeric Analysis of Dynamic Response Under Synchronized Crouching of Pedestrians Laboratory tests and observation of real situations point out that crouching is an activity which can effectively accelerate a footbridge. Especially if a pedestrian bridge is a place of open, mass event. Assumption: Load function was defined after [2], on fig. 6 Action was defined in order to fit two eigenforms. Effective synchronized action can stay no more than 30 seconds. Theoretical load was defined as concentrated and correspondent to the action of 30 pedestrians. Non-linear Newmark time step procedure was proceeded in SOFiSTiK FEM. Fig. 6 Base load function for crouching [2] Theoretical response of the structure under crouching load of 2 x 30 pedestrians acting in opposite phase, with the rate f = 0,542 Hz in the middle of the span is presented in figure 7. The horizontal response of the span in this case can be neglected. uz [mm] 16.000 15.000 14.000 13.000 12.000 11.000 10.000 9.000 8.000 7.000 6.000 5.000 4.000 3.000 2.000 1.000 0.000-1.000-2.000-3.000-4.000-5.000-6.000-7.000-8.000-9.000-10.000-11.000-12.000-13.000-14.000-15.000-16.000-17.000-18.000-19.000-20.000-21.000-22.000-23.000 0.0 2.00 4.00 6.00 8.00 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0 34.0 36.0 38.0 40.0 42.0 44.0 46.0 48.0 50.0 52.0 54.0 56.0 58.0 60.0 62.0 64.0 66.0 68.0 70.0 72.0 74.0 76.0 78.0 80.0 82.0 84.0 sec [sec] Node 216 Fig.7 Vertical displacements of extreme point of carriage way - zmax=22 mm 1069
Footbridge Dynamics Theoretical response of the structure under crouching load of 4 x 30 pedestrians acting in opposite phase with the rate f = 1,45 Hz according to eigen form is presented in figure 8. Vertical acceleration of the extreme point of the carriage way - a max = 7 m/s 2. Horizontal transfer acceleration of the extreme point of the carriage way a max = 1.2 m/s 2 uz [mm] 75.000 70.000 65.000 60.000 55.000 50.000 45.000 Node 322 40.000 35.000 30.000 25.000 20.000 15.000 10.000 5.000 0.000-5.000 0.0 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0 36.0 37.0 38.0 sec [sec] -10.000-15.000-20.000-25.000-30.000-35.000-40.000-45.000-50.000-55.000-60.000-65.000-70.000-75.000-80.000-85.000 3. Dynamic Load Test Fig.8 Vertical displacements of extreme point of carriage way - zmax=85mm Dynamic load test was performed as the second phase of the final testing, just after static proof load and was carried out by KBP Żółtowski in cooperation with Neo Strain Cracow. The complete testing phase was led by Rzeszow University of Technology. Dynamic load test was divided into three phases Phase I identification of self-frequencies. Phase II excitation of the span during walking and running of 100 pedestrians. Phase III excitation of the span during synchronised crouching of 50 pedestrians In Phase I the following eigen frequencies were developed: Eigen form Table 2 Eigen frequencies (estimate and real) Design f[hz] Test f[hz] 1 0.54 0.46 2 1.41 1.53 3 1.66 1.71 4 2.14 2.22 Additionally, logarithmic damping factor was investigated for f 3 and f 4 LDT(f 3 )=~4,8% LDT(f 4 ) =~5,2% 1070
4 th International Conference f 3 i f 4 dominated and as practically placed in the range of pedestrian activity, were pointed as basic frequencies in phase II and phase III of test. Phase II - walking and running on the carriage- ways. During this phase several tests were done with free and synchronised pedestrians. Fig.9 Schema of walking and running during dynamic test The response of the structure was measured under 20, 30, 50 and 100 free and synchronised pedestrians in several combinations, on one or two carriageways of the bridge. The acceleration measurement devices (vertical and horizontal direction) were placed in ¼, ½ and ¾ of total length of the span on both sides (fig.10). Representative results are presented below: Fig.10 Placement of acceleration gages free walking synchronized walking synchronized walking synchronized walking max 100 pedestrians max a V = 0.35 m/s 2 RMS(a)=0.247 m/s 2 max a H = 0.20 m/s 2 RMS(a)=0.14 m/s 2 max 100 pedestrians f=1.9 Hz max a V = 0.6 m/s 2 RMS(a)=0.42 m/s 2 max a H = 0.15 m/s 2 RMS(a)=0.10 m/s 2 max 100 pedestrians f=1.71 Hz - eigen frequency max a v = 1.2 m/s 2 RMS(a)=0.85 m/s 2 max a H = 0.36 m/s 2 RMS(a)=0.25 m/s 2 max 100 pedestrians f=2.22 Hz - eigen frequency max a V = 0.6 m/s 2 RMS(a)=0.42 m/s 2 max a H = 0.1 m/s 2 RMS(a)=0.07 m/s 2 1071
Footbridge Dynamics free run max 100 pedestrians max a V = 1.20 m/s 2 RMS(a)=0.85 m/s 2 max a H = 0.60 m/s 2 RMS(a)=0.42 m/s 2 crouching -max 50 pedestrians f=1.71 Hz - eigen frequency max a V = 1.60 m/s 2 RMS(a)=1.13 m/s 2 max a H = 0.60 m/s 2 RMS(a)=0.42 m/s 2 crouching -max 50 pedestrians f=2.22 Hz - eigen frequency max a V = 3.15 m/s 2 RMS(a)=2.22 m/s 2 max a H = 0.40 m/s 2 RMS(a)=0.28 m/s 2 Crouching in order with the first and second eigen frequency (0,46 Hz and 1,53 Hz) gave no practical effect. 3.1 Safety of the Superstructure Measured vertical displacements under static load test reached maximum value u z =244mm. During the dynamic synchronised crouching test with f=2.22 Hz, we reached maximum acceleration a V =3.15 m/s 2 (synchronised 50 pedestrians). This result lets us estimate the corresponding amplitude of deflection y V =16 mm. This result shows that under synchronised action of 50 pedestrians the structure is safe. Fig. 11 Comfort criteria listed in [3] 1072
th 4 International Conference 3.2 Comfort on the Footbridge Comfort criteria on a footbridge are under continuous discussion, however, a good state of achievements is presented in fig. 11. In the view with the attached comfort criteria the constructed footbridge fulfils the requirements in nearly every case. Under free walking and synchronised walking (f=1.9 Hz) the requirements are reached. During a fully synchronised walk of 100 pedestrians with pace rate f=1.71hz (third eigen frequency) horizontal acceleration was exceeded (0,25 m/s2 > 0,20 m/s2.) During a free run of 100 runners on one side of the bridge horizontal acceleration was exceeded (0,42 m/s2 > 0,20 m/s2.). None of the synchronised walkers and runners realized any discomfort on the footbridge during this phase of the test. Synchronised walking of 100 pedestrians exactly with pace rate of the span and 100 runners on the bridge can be treated as incidental and in this case standard comfort criteria should be omitted. 4. Conclusions Presented paper shows a big influence of dynamic aspect on the design of this unique bridge. Structural elements were dimensioned in respect to global stiffness and mass distribution to reach dynamic requirements. Advanced non-linear time-step procedure was involved to compute dynamic response under purposeful action (crouching). Non-linear behaviour of network hangers with a switch off effect occurred and was the main reason of high reduction of vibrations. This switch off phenomenon was observed during the dynamic load test. Finally, after the successful load test (10.09.2010) the bridge was opened to service and for the last six months there has been no negative sign regarding the dynamic behaviour of the superstructure. Fig. 12 Bridge under dynamic test load 1073
Footbridge Dynamics 5. References [1] BACHMANN H., Lively footbridges a real challenge, Footbridge 2002, Design and dynamic behavior of footbridges, Paris, 20-22.11.2002. [2] ŻÓŁTOWSKI K., Footbridges, numerical approach. Footbridge Vibration Design / eds. E. Caetano; A. Cunha; J. Raoul; W. Hoorpah. - London : CRC Press/Balkema, 2009. - S. 53-70. [3] FLAGA A., PAŃTAK M.; Kryteria komfortu w projektowaniu kładek dla pieszych. Monografia. Projektowanie, budowa i estetyka kładek dla pieszych. Katedra Budowy Mostów i tuneli Politechniki Krakowskiej. Kraków 2003. 1074