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The recent developments in bridge engineering have also affecteddamping capacity of bridge structures. Major sources of dampingin conventional bridgework have been largely eliminated inmodern bridge designs reducing the damping to undesirably lowlevels. As an example, welded joints are extensively usednowadays in modern bridge designs. This has greatly reduced thehysteresis that was provided in riveted or bolted joints in earlierbridges. For cable supported bridges and in particular long spancable-stayed bridges, energy dissipation is very low and is oftennot enough on its own to suppress vibrations. To increase theoverall damping capacity of the bridge structure, one possibleoption is to incorporate external dampers (i.e. discrete dampingdevices such as viscous dampers and tuned mass dampers) into thesystem. Such devices are frequently used today for cable supportedbridges. However, it is not believed that this is always the mosteffective and the most economic solution. Therefore, a great dealof research is needed to investigate the damping capacity ofmodern cable-stayed bridges and to find new alternatives toincrease the overall damping of the bridge structure.
In this paper, the linear dynamic response of a simple two-dimensional cable-stayed bridge model, subjected to a movingvehicle, is studied. Bridge damping, exact cable behavior, andnonlinear geometric effects are considered. This study focuses oninvestigating the influence of vehicle speed, bridge damping,bridge-vehicle interaction, and a tuned mass damper on the bridgedynamic response.
Modern cable-stayed bridges exhibit geometrically nonlinearbehavior, they are very flexible and undergo large displacementsbefore attaining their equilibrium configuration. As an example,due to this inherently nonlinear behavior, conventional linear dead load analysis, which assumes small displacements, is often notapplicable [1].
Cable-stayed bridges consist of cables, pylons and girders (bridgedecks) and are usually modeled using beam and bar elements forthe analysis of the global structural response. To consider thenonlinear behavior of the cables, each cable is usually replaced byone bar element with equivalent cable stiffness. This approach isreferred to as the equivalent modulus approach and has been usedby several investigators, see e.g. [1, 2, 3]. It has been shown in [4]that the equivalent modulus approach results in softer cableresponse as it accounts for the sag effect but does not account forthe stiffening effect due to large displacements. Still, for somecases, e.g. for short span cable-stayed bridges, analysis utilizingthe equivalent modulus approach is often sufficient [3], especiallyin the feasibility design stage. Whereas, long span cable-stayedbridges built today or proposed for future bridges are very flexible,they undergo large displacements, and should therefore beanalyzed taking into account all sources of geometric nonlinearity.Although several investigators studied the behavior of cable-stayedbridges, very few tackled the problem of using cable elements formodeling the cables. See ref. [5, 6] where different cable modelingtechniques are discussed and references to literature dealing withthe analysis and the behavior of cable structures are given.In this paper, an alternative approach is presented where accurateand efficient elements are adopted for the modeling. A beamelement, which includes geometrically nonlinear effects and isderived using a consistent mass formulation, is adopted formodeling the girder and the pylons. Whereas, a two-node cableelement derived using exact analytical expressions for the elasticcatenary, is adopted for modeling the cables. The nonlinear finiteelement method is utilized considering all sources of geometricnonlinearity, i.e. change of cable geometry under different tensionload levels (cable sag effect), change of the bridge geometry due tolarge displacements, and axial force-bending moment interaction inthe bridge deck and pylons (P- δ effect).The adopted beam element, able to resist bending, shear, and axialforces, is developed following the total Lagrangian approach andusing a linear interpolation scheme for the displacementcomponents. This element is chosen because it can handle largedisplacements and shear deformations and because it is simple toformulate the element matrices. This beam element is of minorinterest and, due to space limitation, not discussed here in moredetail. The interested reader is referred to the author’s doctoralthesis, reference [5], where formulation of this beam element ispresented in detail.In the following subsection, the cable element matrices will begiven in the element local coordinate system. Using this approach,each cable may be represented by a single 2-node finite element,which accurately consider the curved geometry of the cable.Despite the fact that this cable modeling technique has beenavailable for many years it has, at least to the author’s knowledge,very seldom been used for analysis of cables in cable-stayedbridges.
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