The Bercy-Tolbiac footbridge in Paris (Feichtinger Architect - RFR Engineers)
Henry BARDSLEY RFR - Paris
Raphaël MENARD RFR - Paris
Bernard VAUDEVILLE RFR - Paris
The future footbridge links the Tolbiac and Bercy development zones in eastern Paris, crossing the Seine as well as the dual carriageways running along its banks. With a central span of 190m without columns in the river, its slender structure is made up of two arches and two catenaries, linked by vertical fingers. In this way two structural systems with similar stiffnesses work in tandem (arches/catenaries and semi-vierendeel trusses). Due to its span and its slenderness, this footbridge is subjected to recently discovered dynamic phenomena under crowd loading, requiring considerable horizontal damping. An original solution is to install viscous dampers at the far edges of the lateral footbridges, which transform them into restraining levers.
Keywords: Footbridge – Slenderness – Arch – Catenary – Semi-Vierendeel – Gerber system – Dynamic crowd loads – Dampers – TMD – Damped lever
In March 1999, the Austrian architect Dietmar Feichtinger and the engineering firm RFR won the international competition, organised by the City of Paris, for the construction of a footbridge over the Seine connecting the quarters of Tolbiac and Bercy in Paris. Construction is to begin at the end of 2003 and is due to be completed at the end of 2005.
Fig.1 Elevation of the structure of the Bercy-Tolbiac footbridge in context The footbridge consists of two levels which cross through each other, permitting pedestrians to access the lower banks of the Seine on one path and the esplanade of the New National Library and Bercy Park on the other path.
The bridge crosses the Seine in one span, the principal supports being set back into the banks of the river. This system allows for a free span of approximately 190 meters. With a total width of 12 meters at the central point, it is intended that the footbridge will not only serve as a means of passage, but will also become a place of leisure, with benches, kiosks, drinks stands, etc installed above the river. On both sides of the Seine, two smaller bridges extend from the elevated path of the main bridge, spanning over the dual carriageways running along the banks.
Fig.2 Plan of the structure of the Bercy-Tolbiac footbridge in context The structural principles of the footbridge are indissociably linked to the architecture. The team of designers worked to fuse the structure into the curves of the bridge deck and to develop long slender proportions. This goal has encouraged a close interaction between architecture and structure, from the definition of the initial concepts through to the development of construction details.
The arch/catenary system
The two crossing decks correspond to the principal components of the structure : a pair of arches and a pair of catenaries.
Fig.3 Schematic structural diagram of the main span of the footbridge The arches and the catenaries transfer the bridge selfweight to the riverside abutments, tracing the curves of the permanent load funiculars. Yet, are these curves sufficiently pronounced to produce the stiffness required for such a bridge ? The proportions of the footbridge are particularly long and slender, contributing th greatly to its elegance : the structural height is less than 6 meters, only 1/30 of the principal span! But this is only an apparent slenderness. The diagram below indicates how the curves of th the arches and the catenaries compound, leading to a real structural ratio of 1/17 , much more usual for steel projects of this type. The overlap of the curves, which cross at th approximately 1/5 of the span, accentuates this effect by creating, to either side of the
central lens, a console which is cantilevered out from the riverside abutment. Significant bending moments are developed in the cantilevers (factored loads of the order of 540MNm).
Fig. 4 Equivalent arch/catenary systems A preliminary approach allows for the axial load in the structure to be estimated using the formula : Average axial force in the arch and the catenary : F = ql² / (8h), with H = hcat + harc, the total structural height, and l= the span. The bridge geometry gives approximately H= 11,05m and l = 185m (l catenary=195m and l arch=174m). For a uniform load corresponding to the self weight of the bridge, q= 40 kN/ml approx, this gives the approximate value : F = 15,5 MN. This approximate value corresponds well with the average effective values of axial force in arch and catenary obtained from calculation, as shown in the graphic below. GSA version 7.4.2 Copyright © Oasys 1999 PasserelledeBercy-Tolbiac, Paris 500 Ouvrages Principales File: A23i00_catène_circle.gwb Element list: G1 to G4 Scale: 1:578.68 Axial Force, Fx: 41517.69 kN/pic.cm 15569.14 kN 11737.70 kN 7906.28 kN 4074.85 kN 243.43 kN -3587.98 kN -7419.42 kN -11250.84 kN Case: A1 :"Poids propre structure + tablier"
Fig. 5 Visualisation of axial tension/compression under permanent loading It is worth noting that the uplift generated by wind action on the deck surfaces is sufficiently weak to avoid any inversion of the tension and compression forces in the arches and the catenaries. This allows the catenaries to be sized for tension only.
The « comb » truss
The static scheme of combined arches/catenaries is no longer sufficient once the funicular profiles of the applied loads diverge from the profile of the deck ; this is
especially the case for the pedestrian live loads which are often not uniformly distributed. To account for the asymmetric part of the applied loads, we superimposed a second static scheme over the first, the two schemes functioning together proportionate to their respective stiffnesses. For this second scheme, we linked the arch and the catenary together with tapered vertical fingers - « obelisks » - built out from the arch at 7-metre step intervals. This creates a « comb » truss in the vertical plane, i.e. a semi-Vierendeel truss in the form of a comb. The longitudinal profile forms a “Gerber”-type system, composed of two cantilevers built out from the abutments and linked by virtual articulations to one central beam in the form of a lens. Although less efficient than a diagonally-braced trellis truss, the «comb » truss has the advantage of not blocking vertical space and not interfering with pedestrian movements. The structure is less obviously expressed and this less immediate image makes the footbridge appear lighter and more transparent. It is worth noting that as a stand-alone structural system, without the additional structural benefit afforded by the curvature of the arches and catenaries, the choice of a semiVierendeel beam for such a large span would not have been reasonable. In this case, however, the lesser stiffness of the semi-vierendeel is an advantage when combined with the arch/catenary system, because it permits the two static systems to function in tandem, without one taking priority over the other.
Fig. 6 The two interlinking static systems 2.3
Stiffness in torsion
The crossing decks of the footbridge are made up of three longitudinal bands, separated by the two « comb » trusses. The outer bands of the deck overhang by 3,5 meters, inducing considerable torsion in the main structure.
Fig.7 Cross-section, central lens
Fig.8 Cross-section, lateral cantilever
To resist this torsion, the upper & lower decks of the central “lens” section are braced in plan and combined with the two vertical trusses into a structural caisson.
On the lateral cantilever sections, only the deck between the arches is braced, and torsional resistance must be achieved not with a caisson but with a U structure. This less efficient configuration is imposed by the absence of a central deck between the catenaries on the right bank. At the junctions between lens and cantilever sections, where the structural height becomes very low, the full width of the central deck is transformed into a closed caisson (6200 x 500 mm) to compensate for the low torsional inertia in this zone. 2.4
Horizontal beams on the underside of the decks
The horizontal beams created on the undersides of the decks to stiffen the bridge in torsion also serve to transmit any horizontally-applied loads (essentially, wind loads) to the riverside abutments. The main chords of these beams are the arches/catenaries, these being linked by horizontal and diagonal braces. 2.5 The abutments The abutments are doubly-loaded : by the compression generated in the arches and by the tension applied by the catenaries. The moment generated by these loads is transferred to the foundations by two inclined « boomerang » props restrained by two rear vertical ties. The abutments on the Left and Right Banks have different configurations depending upon whether or not the boomerang “foot” is monolithic with the joint formed at the arch / boomerang “arm” node. When the boomerang foot is built-in to form a monolithic joint (as on the Right Bank) the abutment is stable under the horizontal loads applied to point H. When the foot is articulated with respect to this node (as on the Left Bank), the tie-beam, the support and the foot form a mechanism that does not resist horizontal loading in F (the rear tie and the boomerang bases are pinned onto the foundations).
Fig. 9 Elevation of the structure and foundations of the West abutment
This configuration allows us to develop statically-determinate support conditions for the main structure. It thus avoids having to design for locked-in stresses induced by thermal loads or by settlement of the supports, and minimises the vertical deformations generated by these loads. The allowable rotation of the boomerang foot at point F is limited in amplitude by a blocking system which locks the articulation when the maximum deformation under « normal » loads is exceeded. The slope given to the boomerang prop in the vertical plane induces an overall tension into the bridge structure; this lightens the arches by reducing the level of compression they are required to resist, although this comes at the price of increased tension in the catenaries.
Fig.10 Axonometric view, calculation model of the footbridge structure
Dynamic analysis of the Bercy-Tolbiac footbridge was carefully considered from the earliest stages of the project. The span and slenderness of the steel frame predict that the bridge will be sensitive to low frequencies. Pedestrian use demands increased attention to comfort requirements. The recently-discovered phenomena of dynamic horizontal coupling between pedestrian movements and footbridge sway, are the object of complementary studies currently being pursued by RFR. These indicate the necessity to include for a high level of horizontal damping. 3.1
Vertical dynamic behaviour
The vertical dynamic pressures generated by the pedestrians are well documented in National and European Standards. Modal analyses (frequency, modal mass, and deflection curves) indicate that the bridge will require reinforced damping of three modes in order to achieve the required comfort criteria. This damping will be obtained by the installation of four Tuned Mass Dampers (TMD). Shown below are the three vertical modes and their corresponding damping device : th
7 mode : fourth symmetrical vertical mode at 1,68Hz Installation of a TMD at mid-span to the lower part of the main footbridge.
Fig.11 Elevation of the mode 7 deformation th
9 mode : second asymmetrical torsion mode at 2,04Hz Installation of two TMD to the belly of the mode 9 deformation. The masses are pushed out towards the exterior in relation to the primary structure.
Fig. 12 Axonometric view of the mode 9 deformation
10 mode: fifth asymmetrical vertical mode at 2,15Hz Installation of one TMD in the cantilever zone on the underside of the principal structure.
Fig. 13 Elevation of the mode 10 deformation 3.2
Horizontal dynamic behaviour
The documentation of horizontal dynamic behaviour of bridges in national and european Standards is not sufficient for design of lightweight longspan pedestrian bridges. The dynamic phenomena that developed at the inaugurations of the Solférino footbridge in Paris and the Millennium Bridge in London illustrate the need for high critical damping ratios for those modes whose frequency falls between 0,4 et 1,3Hz and which develop significant transverse displacements (torsional modes, in particular). Initially we envisaged using horizontal TMD to develop the required damping. But in view of the level of damping required (between 5% and 10%), this solution seemed heavy and over-targeted toward a few specific frequencies. Our current studies are oriented towards the use of internal or external viscous dampers. 3.2.1
Treatment with viscous dampers
This solution consists of making the structure more « viscous » by integrating dampers into the large-deflection zones of the modes that we wish to treat. Unlike the TMD, this solution allows damping of several modes. However, its efficiency varies with the positioning of the dampers within of the structure. A large deformation does not constitute a condition sufficient in and of itself for optimum positioning of the damper connections: in addition, the equivalent rigidity linking the two extremities must also be substantial. The following analysis describes the theory of this « competition » between rigidity and damping. Preliminary calculations We consider that the rheologic model flow+damper in parallel is submitted to a variable instigating force t → F (t ) . This model characterizes the apparent placement of an integrated damper between the two joints of the structure where we wish to improve the damping of the pulse Ω . A joint is the origin of the reference point. The movement of the other extremity is regulated by : Cv + Kx = F (t ) (1) The Fourier transformations are written : ~ x
We are looking for a consistent damping
~ ~ F i.ω.F and ~ v = K + i.ω .C K + i.ω .C
Copt such that at the pulse O, the energy of
damping dissipated in one cycle is at its maximum. This energy is proportional to the quantity
∫ Cv² dt .
By Fourier transformation, we have :
~2 Cω ² F 2 ~ ∫ v² dt ∝ ∫ v dω therefore ∫ Cv² dt ∝ ∫ K ² + C ²ω ² dω ~ Taking F ∝ δ Ω (ω ) . The dissipated energy is CΩ ² therefore proportional to (3) K ² + C ².Ω² This quantity is maximum for C = C opt = K Ω (4) The diagram shows the relationship between the value of the damping constant C and the optimal value of a given pulse. The adjustment of each damper can therefore be optimised in relation to the frequency that we wish to treat. Damping generated in the structure In order to calculate the ratio of critical damping generated by a damper, we establish the following calculation : ξ str , the ratio of critical damping of the structure
M and Ω , modal mass and pure pulse X& , normalized modal speed x& , speed of the lengthening of the damper 2 By applying virtual strengths, we write : 2.ξ str .M .Ω. X& = C opt .x& ² (5) Considering that
X ≡ 1 , we deduce :
ξ str =
Copt .x ² 2.M .Ω
Optimal location of internal dampers We initially examined the installation of viscous dampers parallel to the diagonal braces of the horizontal beams under the decks. We carried out a theoretical analysis, following the approach outlined above, of the efficiency of each potential location. This calculation showed that the optimum locations are situated on the cells 3, 5, and 7 of the horizontal beam linking the two arches, an arrangement which generates a gain in critical damping of the order of 3,3%. Computer analysis using the software package “Algor” allowed us to confirm this value. Although analysis shows that the installation of a structurally-intrinsic damper allows us to substantially increase the ratio of critical damping, the physical application of this device comes up against some difficulties due to the smallness of the relative displacements of the installed
dampers. Furthermore, the required damping constants are very high : it would be necessary to design a kinematic principle capable of increasing displacements whilst at the same time reducing damping forces. 3.2.2
Treatment by external viscous dampers
A more efficient alternative consists of placing the damper between a fixed point independent of the footbridge and a point on the bridge structure that shifts significantly under horizontal modes. The solution described below is still at the preliminary design stage, but is looking promising. The main footbridge is flanked by linking footbridges spanning the dual carriageways situated to either side of the river. In the initial design, the connections between the linking footbridges and the main footbridge are articulated in both the horizontal and the vertical planes. If instead we choose to fix the connection in the horizontal plane between the link bridges and the lateral cantilevers of the main structure, we can mobilise a large additional mass in rotation which translates into a large-amplitude displacement at the outer ends of the link bridges. These two points are ideal locations for external dampers. In physical terms, the viscous dampers would be installed between the outer ends of the link bridges and independent consoles built into the Bercy wall and the Tolbiac bankside, respectively. In this configuration, the plan diagrams below indicate the deformation patterns which we obtain for the most-highly-damped modes :
10 Mode 9’
The graphic below shows the time-transverse displacement relationship (transverse displacement in Oy) of the following nodes : § Node 477 : arch at mid-span ( continuous black line ) § Node 506 : upper tip of the boomerang prop, left bank ( dashed red line) § Node 492 : upper tip of the boomerang prop, right bank ( dotted green line )
The fade-away corresponds to a damping ratio of 10,7% : this value is very close to the theoretical result of 10,6%. 3.3
The future of « restraining lever » footbridges
If current theoretical descriptions of the Millenium-bridge-type phenomena are confirmed, long-span footbridges of the future will have to be designed to achieve very high critical damping ratios for horizontal excitation modes. Our studies show that the principle of « restraining levers » allows high levels of horizontal damping to be attained. This principle consists of creating or mobilising the lateral approaches that are often part of a footbridge design (in our case, the link bridges spanning the dual carriageways) by using them as levers whose rotation can be actively damped.
The future footbridge links the Tolbiac and Bercy development zones in eastern Paris, crossing the Seine as well as the dual carriageways ru...