Closed-form solution for the mode superposition analysis of the vibration in multi-span beam bridges caused by concentrated moving loads

Abstract

In this paper, a closed-form solution for evaluating the dynamical behavior of a general multi-span Bernoulli–Euler beam is derived. The natural frequencies of vibration and corresponding mode shapes are obtained by applying the boundary conditions to the characteristic function of a beam. A Laplace transformation is applied to the governing differential equation which is then solved for each normal mode in the frequency domain. The main contribution of this paper is to provide a closed-form solution for the vibration of continuous stepped beams under constant moving loads. Several numerical examples are included.

Introduction

Shortly after the opening of the new high-speed railway line between Paris and Lyon, it was found that excessive vibration of the bridge deck caused ballast destabilization in a small number of bridges along the line [1], [2]. This destabilization was caused by a resonance phenomenon, i.e. the loading frequency of the train coincided with the natural frequency of the bridge. After a thorough investigation, the European Committee for Standardization (CEN) limited the maximum admissible bridge deck acceleration to 3.5 m/s² for ballasted tracks [2], [3]. The restriction on the maximum admissible deck acceleration requires detailed dynamical analyses, which are usually performed by using finite element models. This type of analysis however tends to be time consuming. Therefore, according to the authors opinion, there is a need, especially in the early design stages, for a robust but faster analysis tool. Such a tool must be based on an analytical or semi-analytical closed form solution to the problem.

Over the past 60 years, a number of researchers have studied the moving load problem for continuous beams with the aim of finding a simple model that can describe the main characteristics of their behavior. The vibration of a beam with two equal spans under a constant moving force was first solved by Ayre et al. [4]. Their solution was based on describing the moving load as a series of pulsating forces. This solution was later extended by Saibel and Lee [5] who introduced a method that is applicable to a more general case, even though the solution is only presented for a two-span beam. Both authors assume that the beam is undamped. In as far as these early studies are concerned, an excellent state of the art review was presented by Fryba [6] in 1972. In the cited reference, one chapter presents a solution to the moving load problem for a simply supported beam using a Laplace–Carson integral transformation. Hamada [7] solved the moving load problem for a simply supported beam by a double Laplace transformation. Both free vibration and the governing differential equations were solved by this method. Dmitriev [8] studied the dynamic behavior of a three-span beam under a moving load. His work was later extended to a two-span beam with elastic central support [9] and a multi-span beam [10]. These solutions are nevertheless limited to undamped beams with the same span properties, i.e., span length, mass per unit length, and flexural stiffness. Vibrations of an infinite continuous beam subjected to a moving force have been studied by Cai et al. [11]. After the governing equation is decoupled using the U-transformation, the time response was obtained by Duhamel’s integral. Wang [12] investigated the dynamic response of a continuous undamped beam considering shear deformation by using Timoshenko’s beam theory. The structural response is obtained as an integral, solved by a numerical procedure. Zheng et al. [13] calculated the dynamic response of an undamped continuous beam loaded with a moving load using a modified beam vibration function as an assumed mode. This method was later extended by considering the load as a system with two degrees of freedom [14]. Dugush and Eisenberger [15] used an infinite polynomial series for the mode shapes to obtain the exact solution for non-uniform sections. Several authors have calculated the time response with a recurrence formula, similar to a numerical procedure [13], [14], [15]. More recently Martinez-Castro et al. [16] presented a semi-analytical solution for damped non-uniform continuous beams. The natural frequencies and mode shapes are calculated with conventional finite element methods by approximating the shape function with a cubic Hermitian polynomial. Xu et al. [17] studied a continuous beam with different boundary conditions and semi-rigid connections between spans. In 2010, Salvo et al. [18] proposed an alternative method to determine the mode shapes and natural frequencies based on a component mode synthesis (CMS). In addition, [19], [20], [21], [22], [23], [24] should also be mentioned, since their work also treats this problem.

However, all the above mentioned methods suffer from limitations, which mainly fall into two categories: either the model is oversimplified, e.g., by neglecting damping, or the governing differential equations are solved in a recursive or approximate manner. In this context, the present study aims at removing the above mentioned drawbacks associated with such simplified solutions by deriving an exact closed form solution to calculate the displacements, velocities, and accelerations of a stepped continuous girder bridge supported by elastic springs. The governing differential equation is first transformed into a modal equation assuming it to be a linear combination of normal modes. The natural frequencies and normal modes are then determined by substituting the boundary conditions into the characteristic function of the beam. The modal equation is then solved in the frequency domain using a Laplace transformation. This part essentially follows a procedure proposed by Hayashikawa and Watanabe [25]. However, instead of solving the equations by means of a numerical procedure, a closed-form solution is derived and presented in this paper. As far as the authors know, this is the first time an exact closed-form solution has been derived for the moving load problem in a general case which considers several spans with different ratios, stepped sections within spans, damping and elastic boundary conditions. Unlike previous publications, the main objective of this paper is to investigate accelerations rather than displacements.