Development of an efficient finite element model for the dynamic analysis of the train-bridge interaction

The design of high-speed railway bridges comprises a set of demands, from safety and serviceability aspects, to new types of equipment and construction solutions. In order to perform an accurate and realistic evaluation of the corresponding dynamic behavior, adequate analysis tools that take into account the complexity of the train-bridge system are required. These computational tools must be based on efficient algorithms to allow for the completion of detailed dynamic analyses in a reasonable amount of time. The classical methods of analysis may be unsatisfactory in the evaluation of the dynamic effects of the train-bridge system and fully assessment of the structural safety, track safety and passenger comfort. A direct and versatile technique for the simulation of the train-bridge interaction was implemented in the FEMIX code, which is a general purpose finite element computer program. The presented case study is an application of the proposed formulation, which proved to be very accurate and efficient. 2 HHT METHOD WITH TRAIN-BRIDGE INTERACTION A simple example is used to introduce the types of degrees of freedom that are considered in the formulation of the vehicle-structure interaction in the context of a time step of the Hilber-Hughes-Taylor method (see Fig. 1). On the right, a simply supported beam with two spans (B1 and B2) is subjected to the contact of a vehicle, shown on the left. The structure of the vehicle is also composed of two beams (B3 and B4). Nodes 7, 8 and 9 are internal points of the beam B1. The location of these nodes may change between time steps, depending on the position of the vehicle. Eventual gaps between both structures (gi) can be easily considered in the compatibility equations, as will be shown later. Figure 1. Vehicle and structure: beams and nodal points. In each nodal point two degrees of freedom are considered (vertical displacement and rotation). Figure 2 shows the generalized displacements in nodal points (1 to 12), the generalized displacements of the contact points of the structure (13, 14 and 15), the interaction forces in the vehicle (X7, X9 and X11) and the interaction forces in the structure (Y13, Y14 and Y15). The interaction only involves the translational degrees of freedom. Figure 2. Vehicle and structure: degrees of freedom and interactions forces. The following classification of the degrees of freedom is considered: − F – free; − X – interaction (vehicle); − P – prescribed; − Y – interaction (structure). This classification is used later in this section. In the context of the Hilber-Hughes-Taylor method (HHT), the dynamic equilibrium equation that involves the degrees of freedom in nodal points (1 to 12) is the following ( ) ( ) ( ) p c p c p c c α α α α α α F F u K u K u C u C u M − + = − + + − + + 1 1 1 ɺ ɺ ɺ ɺ (1) In this equation M is the mass matrix, C is the damping matrix, K is the stiffness matrix, F are the applied generalized forces, u are the generalized displacements and α is the main parameter of the HHT method. When α = 0 the HHT method reduces to the Newmark method, and for other values of the parameter α, numerical energy dissipation is introduced in the higher modes. The superscript c indicates the current time step (t + ∆t) and the superscript p indicates the previous one (t). According to Figure 2 and to the classification indicated above, the F type degrees of freedom are the following: 2, 4, 6, 8, 10 and 12. The X type degrees of freedom correspond to the 1 7 8 9 3 2 6 5 4