آنالیز اثر دینامیکی طول پل های کابلی تحت بارهای در حال حرکت

The aim of this paper is to investigate the dynamic response of long span cable-stayed bridges subjected to moving loads. The analysis is based on a continuum model of the bridge, in which the stay spacing is assumed to be small in comparison with the whole bridge length. As a consequence, the interaction forces between the girder, towers and cable system are described by means of continuous distributed functions. A direct integration method to solve the governing equilibrium equations has been utilized and numerical results, in the dimensionless context, have been proposed to quantify the dynamic impact factors for displacement and stress variables. Moreover, in order to evaluate, numerically, the influence of coupling effects between bridge deformations and moving loads, the analysis focuses attention on the usually neglected non-standard terms related to both centripetal and Coriolis forces. Finally, results are presented with respect to eccentric loads, which introduce both flexural and torsional deformation modes. Sensitivity analyses have been proposed in terms of dynamic impact factors, emphasizing the effects produced by the external mass of the moving system and the influence of both “A” and “H” shaped tower typologies on the dynamic behaviour of the bridge.

Cable-stayed systems have been employed, frequently, to overcome long spans, because of their economic and structural advantages. Moreover, improvements in the use of lightweight and high strength materials have been proposed in different applications, and, consequently, more slender girder cross sections have been adopted. As a result, the external loads have become comparable with those involved by the bridge self-weight ones and an accurate description of the effects of the moving loads is needed to properly evaluate dynamic bridge behaviour. At the same time, new developments in rapid transportation systems make it possible to increase the allowable speed range and traffic load capacity; consequently, the moving system can greatly influence the dynamic bridge vibration, by means of non-standard excitation modes. To this end, investigation is needed to quantify the effects produced by the inertial forces of the moving system on the bridge vibration. The extension of the moving load problem to long span cable-supported bridges requires a consistent approach, appropriately formulated, in order to fully characterize the bridge kinematics and train–girder interaction. In the literature, several studies have been developed, which analyse dynamic bridge behaviour with respect to different assumptions and frameworks. In particular, Fryba and Timoshenko [1] and [2], provided a comprehensive treatment concerning primarily the dynamic response of simply supported girder structures travelled by vehicles, and analytical as well as numerical solutions for some specific problems have been presented. During the last few decades, with advances in high performance computers and computational technologies, more realistic modelling of the dynamic interaction between a moving system and bridge vibration has become feasible. In particular, Yang et al. [3] presented a closed-form solution for the dynamic response of simple beams subjected to a series of moving loads at high speeds, in which the phenomena of resonance and cancellation have been identified. Moreover, Lei and Noda [4] proposed a dynamic computational model for the vehicle and track coupling system including girder profile irregularity by the finite element method, whereas additional references to the influence of AASHTO live-load deflection criteria on the vibration in a railway track under moving vehicles can be found in [5], [6] and [7]. With reference to cable-stayed bridges, in order to evaluate the amplification effects produced by the moving system, different investigations have been proposed. In particular, Au et al. [8] and [9] investigated the dynamic impact factors of cable-stayed bridges under railway traffic using various vehicle models, evaluating the effects produced by random road surface roughness and long term deflection of the concrete deck. An efficient numerical modelling has been developed by Yang and Fonder [10] to analyse the dynamic behaviour of cable-stayed bridges subject to railway loads, taking into account nonlinearities involved in the cable system. Dynamic interaction of cable-stayed bridges with reference to railway loads has been investigated in [11], in which strategies to reduce the multiple resonant peaks of cable-stayed bridges that may be excited by high-speed trains have been proposed for a small length bridge structure. Finally, a computational model and a parametric study have been proposed in [12] to investigate bridge vibration produced by vehicular traffic loads. The literature referred to above investigates dynamic bridge behaviour properly taking into account the effects of interaction between bridge vibration and the moving system. However, only a few studies have concentrated on the dynamic responses of long span bridges. This paper, therefore, focuses on the dynamic behaviour of long span cable-stayed bridges, evaluating the effects produced by the moving system on the dynamic bridge behaviour. In particular, the main aims of this paper are to propose a parametric study in a dimensionless context, which describes the relationship between dynamic amplification factors and moving loads and bridge characteristics. The structural model is based on a continuum approach, which has been widely used in the literature to analyse long span bridges [13], [14] and [15]. In particular, Meisenholder and Weidlinger [13] have schematized bridge structures as an elastic beam resting on an elastic foundation, whose stiffness is strictly connected to the geometrical and stiffness properties of the stays. Moreover, extended models which generalize the bridge kinematics have been proposed in [14] and [15], in which the stay spacing is assumed to be small in comparison with the central bridge span. As a result, the interaction forces between the cable system and the girder can be assumed as continuous functions distributed over the whole girder length. The accuracy of the continuum approach has been validated in previous works developed in both static and dynamic frameworks, through comparisons with numerical results obtained by using a finite element model of the discrete cable system bridge [14], [15] and [16]. In the present paper, the bridge kinematics and the inertial forces have been considered in a tridimensional context, in which both in-plane and out-of-plane deformation modes have been accounted for. Cable-stayed bridges based on both “H” and “A” shaped typologies with a double layer of stays have been considered. However, cable-stayed bridges with one central layer of stays, especially for eccentric railway bridges, are characterized by high deformability, and difficulties verifying the design rules on maximum displacements occur frequently. In particular, the girder torsional stiffness needs to be significantly improved with respect to those involved for “H” and “A” shaped typologies, because contributions arising from the cable system are practically negligible. As a matter of fact, torsional analysis carried out for typical concrete or steel girder cross sections shows that in order to limit torsional rotation to reasonable values (i.e. below 0.02), the maximum allowable central length must be approximately equal to 400 m [16]. The equations of motion for the vehicle-track-bridge element are derived by means of the Hamilton principle. Subsequently, the boundary value problem, due to the equilibrium equations, was solved, numerically, by means of a finite difference scheme based on θθ-family methods, in which proper interpolation functions on both spatial and time domains were adopted to obtain stable and accurate results. A parametric study in a dimensionless context has been analysed by means of numerical results, in terms of typical kinematic and stress bridge variables for both in-plane and eccentric loading conditions. In particular, results are proposed to investigate the effects of moving the system description with reference to non-standard forces, usually neglected in conventional dynamic analyses, i.e. Coriolis and centripetal accelerations. Finally, the influence on the dynamic bridge behaviour of pylon typology with reference to both “A” and “H” shapes has been analysed,and comparisons in terms of both moving loads and tower characteristics have been proposed.

Long span bridges under moving loads have been analysed for both flexural and torsional deformation modes, in terms of dynamic impact factors for typical kinematic and stress variables of the bridge. The effects of the inertial description of the moving system on the dynamic bridge behaviour have been investigated, by means of a parametric study developed in terms of both moving loads and bridge characteristics. The influence of the inertial forces are considerable, while those corresponding to non-standard contributions arising from Coriolis and centripetal accelerations determine the major amplifications, mainly at high speeds of the moving system. The inertial effects of the moving system have been discussed with respect to typical geometrical and stiffness parameters of the bridge, emphasizing the amplification effects produced by the inertial forces of the moving system. For eccentric loads, sensitivity analyses have been developed in terms of dynamic impact factors and maximum normalized displacements with respect to both “A” and “H” shaped tower typologies. In the framework of the “A” shaped tower typologies, the coupling behaviour between torsional and transversal flexural deformations has been discussed. In particular, the influence of the transverse displacements has been investigated, by means of sensitivity analyses. This establishes that, for an in-plane loading condition, the effect of transverse deformability on the dynamic behaviour of the bridge is practically negligible. Moreover, numerical results have shown that, in comparisons with the “H” shaped tower topology, the “A” shaped ones, even if having greater dynamic amplification factors, are characterized by enhanced stiffness properties, which are able to efficiently reduce torsional bridge deformation. The investigation is developed in terms of the main dimensionless parameters related to both geometric and stiffness properties of the bridge. As a result, a parametric study may be useful in the design procedure since the dynamic impact factors for typical deformation and stress variables can be determined in advance.