Shock and Vibration

Volume 2016, Article ID 4070627, 12 pages

http://dx.doi.org/10.1155/2016/4070627

## Application of Volterra Integral Equations in Dynamics of Multispan Uniform Continuous Beams Subjected to a Moving Load

The Faculty of Environmental Engineering and Geodesy, Wrocław University of Environmental and Life Science, Grunwaldzka 55, 50-365 Wrocław, Poland

Received 30 March 2016; Revised 6 September 2016; Accepted 8 September 2016

Academic Editor: Salvatore Russo

Copyright © 2016 Filip Zakęś and Paweł Śniady. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The dynamic behavior of multispan uniform continuous beam arbitrarily supported on its edges subjected to various types of moving noninertial loads is studied. Problem is solved by replacing a multispan structure with a single-span beam loaded with a given moving load and redundant forces situated in the positions of the intermediate supports. Redundant forces are obtained by solving Volterra integral equations of the first or the second order (depending on the stiffness of the intermediate supports) which are consistent deformation equations corresponding to each redundant. Solutions for the beam arbitrarily supported on its edges (pinned or fixed) due to a moving concentrated force and moving distributed load are given. The difficulty of solving Volterra integral equations analytically is bypassed by proposing a simple numerical procedure. Numerical examples of two- and three-span beam have been included in order to show the efficiency of the presented method.

#### 1. Introduction

Many authors have considered the problem of vibrations in structural and mechanical engineering resulting from the moving load, because of both being interesting from the theoretical point of view and having a significant importance for the practice. This problem occurs in dynamics of bridges, roadways, railways, and runways as well as missiles, aircrafts, and other structures. Various types of structures and girders like beams, plates, shells, and frames have been considered. Also various models of moving loads have been assumed [1]. Both deterministic and stochastic approaches have been presented [2–4].

In most studies a single-span girder like a string, a beam, a plate, or a shell has been considered. The solution of the response of a finite, single-span beam subjected to a force moving with a constant velocity has a form of an infinite series and has been presented in many papers. Original solutions in a closed form for the aperiodic vibration of the finite, simply supported Euler-Bernoulli beam, Timoshenko beam, and a sandwich beam are given in the papers [5–7]. Also more complex systems like a double-string, a double-beam, or a suspension bridge have been considered as single-span girders [8–12]. An important and interesting problem is the vibrations of a multispan beam caused by a moving load. There are many structures, for example, bridges, which are multispan. There are not so many papers focused on the dynamic response problem of a multispan beam due to a moving load [13–31]. The vibrations of a multispan Bernoulli-Euler beam with an arbitrary geometry in each span subjected to moving forces [13–19], or moving masses [24, 25], or moving oscillators [26] have been considered. Also the vibrations of a multispan Timoshenko beam due to moving load have been considered [27–29]. Vibrations of multispan sandwich or composite beams are considered in the papers [30, 31]. The solutions for the vibration of a frame caused by a moving force are given in the paper [32].

In this paper the dynamic behavior of Euler-Bernoulli multispan uniform continuous beam system traversed by a moving load is analyzed. We combine analytical and numerical procedures to present a solution for the case of a beam traversed by a constant force moving with the constant velocity. It is assumed that the stiffness and the mass of the beam in every span are the same but the lengths of the spans can be different. The problem is solved similar to the static force method but instead of a set of algebraic equations we have to solve a set of Volterra integral equations (first order when the beam rests on supports of infinite stiffness or the second order when the beam rests on elastic supports). It is difficult to solve these Volterra integral equations in analytical way; for this reason they should be solved numerically. The primary structure (primary beam) is an arbitrarily supported single-span beam. For this reason, in order to find the solution for multispan continuous beams using a set of the Volterra integral equations, in the first step the dynamic response of a finite, single-span beam subjected to a moving load and stationary point forces is considered. The presented algorithm is used to determine the vibrations of two- and three-span beams. The correctness of the algorithm has been tested using Finite Difference Method.

#### 2. Vibrations of an Arbitrarily Supported Single-Span Beam under a Moving Load

Let us consider Euler-Bernoulli beam element of constant flexural rigidity and constant mass per unit length subjected to a dynamic load . Equation of motion describing undamped vibrations has the form Let us assume that the beam is of finite length and has pinned or fixed supports on both ends. The beam rests also on arbitrarily located intermediate point supports and is subjected to a load moving with constant velocity (see Figure 1).