Dynamic Interaction of Bridges and High-Speed Trains

Dynamic interaction of railway bridges and rolling stocks has been the subject of numerous studies for a rather long time [1-20]. The framework of railway bridges experiences significant loads from high-speed trains. Therefore, new methods for studying the interaction between high-speed trains and railway bridges are constantly developed in many countries [10-19]. The method proposed in [9, 16, 18, 19] for beams, rod and combined systems, bridge girders, and railway structures takes into account any required number of shapes in the deflection decomposition and has the resolving system of equations with unconditionally stable integration scheme and minimum number of unknowns (dynamic reaction of the wheels). The bridge dynamics problems of large dimension [16-20] are usually solved by the method of direct integration of equations (with the use of beam boundary elements), but the method proposed here is based on the use of the results of the modal analysis of the structure, which is very convenient, with the capabilities of the modern computational complexes taken into account, for determining the set of natural modes and frequencies of vibrations of any spatial linearly deformable bridge structures. Just as in [16-20] for problems with moving load, the method permits obtaining the resolving system of equations with minimum number of unknowns for a high degree of spatial digitation of the bridge framework and any necessary number of natural modes. Moreover, the step procedures proposed in [19, 20] to solve the differential equations of mechanical system vibrations permit using the car model as a moving load in the same way in the linear and nonlinear statements. In the proposed method, there are in fact two objects; namely, the set of bridge spans and the rolling stock (stocks) are considered as two super-elements (two substructures) interacting with each other if one of them is moving. The proposed method permits determining the stress-strain state of the bridge and the rolling stock traditionally after determining the dynamic pressures exerted by the wheeled carts on the bridge. The modern approaches to calculating the bridge-rolling stock interaction [10-17] are based on the wide use of computers with application of the finite element method and method of stepwise integration of equations describing the system motion. For example, in [10, 13] the iteration approach is used in problems of large dimension when the rolling stock-bridge system is decomposed. In [12], the explicit scheme of integration of differential equations is used to compensate for the large dimension of the problem. In [14], the beam finite elements are used to analyze a single-span bridge of box section in the span structure modeling. In [14], at the stage of time-digitation (by the Newton method), after the "rolling stock-bridge" system is decomposed into subsystems, the constraint reactions are omitted but the large dimension of the problem is preserved. Thus, the problem of large dimension in problems of bridge dynamics is often solved approximately with a certain accuracy of the solution.