Worldwide, high-speed railway (HSR) lines are expanding, while the ratio of bridges they are made up of is increasing. At the same time, the unprecedented speeds of HSR trains intensify the dynamic interaction between trains and bridges. As a result, dynamic phenomena (e.g., resonance) may develop, which jeopardize the safety and comfort of rail transport. This realization motivated a proliferating research field on the dynamic vehicle–bridge interaction (VBI) phenomenon. Despite the eminent progress in the VBI field during the last decades, a consistent theoretical foundation of the dynamics between vehicles and bridges is still lacking. On the other hand, there is still a need for robust schemes to numerically treat VBI. This study aims to address such shortcomings, first, by investig...[ Read more ]

Worldwide, high-speed railway (HSR) lines are expanding, while the ratio of bridges they are made up of is increasing. At the same time, the unprecedented speeds of HSR trains intensify the dynamic interaction between trains and bridges. As a result, dynamic phenomena (e.g., resonance) may develop, which jeopardize the safety and comfort of rail transport. This realization motivated a proliferating research field on the dynamic vehicle–bridge interaction (VBI) phenomenon. Despite the eminent progress in the VBI field during the last decades, a consistent theoretical foundation of the dynamics between vehicles and bridges is still lacking. On the other hand, there is still a need for robust schemes to numerically treat VBI. This study aims to address such shortcomings, first, by investigating analytically the physical mechanisms of VBI and, following, by developing a theoretically sound numerical algorithm to solve the VBI problem.

As a first approach, the study examines analytically the dynamic response of a single-degree of freedom (SDOF) vehicle–SDOF bridge configuration. Through this analysis it reveals that the impedance ratio—defined as the ratio of the vehicle’s damping to bridge’s mechanical impedance—is a dominant coupling parameter of VBI, in addition to the well-known vehicle-to-bridge stiffness ratio. Following, it derives explicit analytical expressions for the constituent mechanisms of VBI on the mechanical system of the bridge. These mechanisms break down to (a) an additional damping term, (b) an additional stiffness term, and (c) a modified loading term. Focusing on the additional damping term, the study offers simplified formulas to estimate the additional damping ratio for common bridge types, such as simply supported and continuous bridges. The proposed formulas appear to be more accurate compared with currently available methods to estimate the additional damping ratio of bridges due to VBI, such as the Additional Damping Method of Eurocode.

With reference to railway bridges, the study proposes a Modified Bridge System (MBS) method to decouple the VBI problem when the vehicle-to-bridge stiffness ratio is small. The MBS approach solves a modified bridge system, independent of the traversing vehicle. The response of the traversing vehicle can then be estimated from the calculated response of the bridge. The MBS method is applicable to bridges that can be described as SDOF systems (e.g., simply supported bridges), and outperforms other decoupling approaches, such as the moving load approximation and the dynamic analysis proposed by Eurocode. At the same time, compared to the solution of the coupled vehicle-bridge system, it is equally accurate but computationally more efficient. However, it is limited by the single-mode representation of both vehicle and bridge.

To this end, the study develops an Extended Modified Bridge System (EMBS) method. This constitutes an extension of the MBS method to multi-degree of freedom (MDOF) vehicle–MDOF bridge systems. The examined MDOF vehicle–MDOF bridge configuration accounts for higher modes of both vehicle and bridge, as well as for the mass of the wheels in contact with the bridge. Thus, it is representative of a wide class of practical train–bridge systems. As a result, EMBS method is appropriate for involved bridge configurations, such as continuous and arch bridges. At the same time, it allows the accurate estimation of bridge-deck acceleration, which is an important serviceability limit state. Moreover, the MDOF description of the constituent effects of VBI brings forward important dynamic phenomena, which are overlooked by the corresponding SDOF description. For example, under certain conditions, the time-varying additional damping matrix of bridges can also obtain negative values, which implies influx instead of dissipation of energy. Further, the mass of the vehicle’s wheels has a prominent effect on bridge acceleration, especially in the presence of irregularities.

The second part of this study focuses on the development of a robust and cost-effective numerical analysis scheme to treat VBI, the localized Lagrange multipliers algorithm. As a first approach, the proposed algorithm partitions the coupled vehicle–bridge system by introducing artificial auxiliary contact points between the vehicle’s wheels and the bridge deck elements in contact. The artificial points allow the formulation of two sets of kinematic constraints and two sets of contact forces (i.e., localized Lagrange multipliers) between the vehicle and the bridge, which enable the partitioned, non-iterative dynamic analysis of the two subsystems. The distinct feature of the localized Lagrange multipliers algorithm is that it decreases significantly the computational cost of the VBI analysis, especially for large vehicle–bridge systems. However, it solves a system of index-3 differential-algebraic equations (DAEs), typical of algorithms to solve constrained mechanical systems, thus requires stabilization of the involved constraints.

Therefore, the study introduces a Dynamic Partitioning Method (DPM) that preserves the stability of the constraints during time-integration. A characteristic difference of DPM compared with the localized Lagrange multipliers algorithm is the introduction of auxiliary contact bodies, instead of merely points, between the vehicle’s wheels and the sustaining bridge. This allows the assignment of proper mass, damping, and stiffness properties to the involved constrains. These properties are estimated based on a consistent application of Newton’s law of motion to mechanical systems subjected to bilateral constraints. This leads to a dynamic representation of motion constraints and associated Lagrange multipliers. As a result, equations of motion and constraint equations yield a set of second-order ordinary differential equations (ODEs), which avoid numerical drifts and instabilities associated with DAEs. Lastly, for the time-integration of the ODEs system, DPM employs a cost-effective alternative augmented Lagrangian formulation, thus it treats VBI both accurately and efficiently.

Author’s keywords: vehicle–bridge interaction, high-speed railway, railway bridge, decoupled analysis, additional damping method, partitioned algorithm, dynamic Lagrange multipliers, numerical stability