1. Introduction

Analysis of the effects of moving loads on bridge structures was motivated by the development of rail transport in the two last centuries, which necessitated the construction of many bridges. The busy bridge transport operation eventually resulted in failures, e.g., the collapse of Chester rail bridge in 1947, Takoma highway bridge in 1940, etc. The first theoretical studies of dynamic bridge response, idealized as an elastic beam of finite length with a moving mass point, were presented in 1849 by Willis [1] and Stokes [2] and later in 1896 by Zimmerman [3]. The moving of massless force across a beam was analyzed by Krylov [4] and Timoshenko [5], who also simultaneously solved the problem of force moving across a mass-beam at constant speed. The total knowledge of the problem from that period was summarized by Inglis [6]. Nowadays, similar bridges problems are solved by numerical finite element methods via modal co-ordinate analysis of structures subjected to moving loads, e.g., Rao [7]. While the problem of rail bridge vibration has been investigated intensively since the second half of the last century, serious attempts to solve the problems of highway bridge vibration date from

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

the middle of this century. The first report on this problem was published in 1931 by the American Society of Civil Engineers—ASCE [8] after which significant advances were made using analogue and digital computers, see also Biggs et al. [9], Looney [10], Huang and Valetsos [11], Tung et al. [12], Chaallal and Shahawi [13]. Czechoslovakia (till 1993), relevant studies were performed by Koloušek [14], Frýba [15], Baťa [16], Benčat [17], and others.

negligible compared with the mass of the beam. This basic case of dynamic bridge response was solved by authors, e.g., [4, 6, 13, 15] and others. The vibration caused by a force moving across an elastic Bernoulli-Euler's beam (Figure 1) with viscous damping, is described by the

is the static deflection at mid-span of the beam. The circular frequency of the damped beam

ð1Þ

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Bridges Subjected to Dynamic Loading http://dx.doi.org/10.5772/intechopen.73193

ð2Þ

ð3Þ

ð4Þ

equation

where

vibration, with light damping, is

and with heavy damping is

Figure 1. Simple beam subjected to a moving load.

with appropriate boundary and initial conditions.

The state of oscillation may be expressed as follows:

In the majority of studies, the bridge is considered as a one-dimensional beam, for which the differential equations of motion have been solved by numerical methods. The application of advanced calculation methods (finite element methods—FEM and other relevant numerical methods) enables two- and three-dimensional simulation models of bridge vibration to be solved. In the 1970s, Ting et al. [18] proposed an algorithm for solving this problem, based on an integral formulation of bridge vibration, which took into account the relations of kinematic bond of the moving vehicle-and-bridge interaction. The application of the above-mentioned methods in many cases was successful and led to the introduction of design standards and methods of assessing bridge structures. From a historical point of view, these solutions represent a gradual development in the understanding of bridge dynamic response, due to moving vehicles and their interactions.

The solution of bridge service life and reliability problems, as influenced by bridge vibration, which is mostly of random character, contributes to the complexity of this problem. However, in spite of all the complications of bridge vibration and the numerous parameters incorporated in regulations and standards in many countries, the natural frequencies and corresponding modes of vibration, the dynamic coefficient (dynamic increase of stress or deformation) and the damping are the basic bridge vibration characteristics, which can be verified by in situ experimental tests and monitoring.

Presently for evaluation of dynamic response and projected parameters, new and existing bridges are utilizing numerical and experimental bridges dynamic analysis. Full-scale bridges dynamic testing and monitoring give relevant information for projecting and assessment of real bridge behaviors [16–23]. This information consists of observed quantities obtained by experimental tests, theoretical analysis, and numerical computation and their comparison. Nowadays, the important role in the control of the bridge structures with bridge dynamic parameters (relative change of eigen-frequencies, damping parameters, fatigue parameters, vibration effective amplitudes value in time histories, etc.) plays monitoring of the structural parameters during normal bridge traffic on. Some results from bridge forced vibration tests (vibration is artificially induced (e.g., during the dynamic loading tests—DLT, etc.)) and also from bridge monitoring ambient vibration tests (input excitation is not under the control of the test engineer) are also used.
