Comparison of Two Types of Deflection Functions for Analysing the Responses of the Rail and the Bridge under Static or Moving Vehicles

Abstract

The interaction model consisting of a train, a two-layer railway track, and a railway bridge is presented for an analysis of the responses of the rail and the bridge, and two types of deflection functions are compared. One type is the shape functions of a beam element described by the cubic Hermitian interpolation functions in the finite-element method (FEM), and the other is the deflection of a beam described by the superimposing modes in the modal analysis method (MAM). In the present model, each vehicle is modelled as a two-wheel mass-spring-damping system with four degrees of freedom (DOFs); the track is modelled as a discrete supported undamped uniform Bernoulli-Euler beam with hinged-hinged ends representing rails, discrete lumped masses representing sleepers, and discrete massless springs and dashpots representing the characteristics of pads and ballasts; and the bridge deck is modelled as a simply supported uniform Bernoulli-Euler beam. Two different equations of motion for the present model with two types of deflection functions for the rail and the bridge deck are derived from the energy principle. These equations can be easily degenerated into the equations of either a train-track or train-bridge interaction system. Numerical results show that the difference between the two types of deflection functions to analyse the responses of the bridge is very small; and FEM can, yet MAM cannot, give an acceptable result if the bending moment of the rail is taken into account.