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A THREE-DIMENSIONAL RAILÂBRIDGE COUPLING ELEMENT OF UNEQUAL LENGTHS IS PRESENTED IN THIS PAPER, IN WHICH THE LENGTH OF THE RAIL ELEMENT IS SHORTER THAN THAT OF THE BRIDGE ELEMENT, TO INVESTIGATE THE SPATIAL DYNAMIC RESPONSES OF A TRAINÂTRACKÂBRIDGE INTERACTION SYSTEM. THE FORMULATION OF STIFFNESS AND DAMPING MATRICES FOR THE FASTENER, BALLAST, AND BEARING, AS WELL AS THE THREE-DIMENSIONAL EQUATIONS OF MOTION IN MATRIX FORM FOR A TRAINÂTRACKÂBRIDGE INTER-ACTION SYSTEM WITH THE PROPOSED ELEMENT, ARE DERIVED IN DETAIL BY USING ENERGY PRINCIPLE. TWO EXAMPLES WITH A SEVEN-SPAN CONTINU-OUS BEAM BRIDGE ARE SHOWN. THE FIRST EXAMPLE IS TO INVESTIGATE THE INFLUENCE OF THE EFFICIENCY AND ACCURACY OF THE LENGTHS OF THE RAIL AND BRIDGE ELEMENTS ON THE SPATIAL DYNAMIC RESPONSES OF THE TRAINÂTRACKÂBRIDGE INTERACTION SYSTEM, WHILE THE OTHER IS TO INVES-TIGATE THE INFLUENCE OF TWO TYPES OF TRACK MODELS ON THE DYNAMIC RESPONSES OF THE SYSTEM. THE RESULTS SHOW THAT: (1) THE PROPOSED RAILÂBRIDGE COUPLING ELEMENT CAN NOT ONLY HELP TO SAVE CALCULATION TIME BUT ALSO GIVE SATISFACTORY RESULTS IN INVESTIGATING THE SPATIAL DYNAMIC RESPONSES OF A TRAINÂTRACKÂBRIDGE INTERACTION SYSTEM, (2) THE DOUBLE-LAYER TRACK MODEL IS MORE ACCURATE COMPARED WITH THE SINGLE-LAYER TRACK MODEL ESPECIALLY FOR THE VIBRATIONS OF BRIDGE AND RAIL.

Three-Dimensional Rail-Bridge Coupling Element of Unequal Lengths for Analyzing Train-Track-Bridge Interaction System