Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 835460, 10 pages

http://dx.doi.org/10.1155/2015/835460

## In-Plane Vibration Response of the Periodic Viaduct on Saturated Soil under Rayleigh Surface Wave

^{1}School of Civil Engineering and Architecture, Nanchang University, Nanchang, Jiangxi 330031, China^{2}Nanchang Institute of Technology, Nanchang, Jiangxi 330099, China

Received 10 April 2015; Accepted 1 July 2015

Academic Editor: Xiaobo Qu

Copyright © 2015 Hai-yan Ju and Ming-fu Fu. 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

In this study, the in-plane vibration response of the periodic viaduct on saturated soil under Rayleigh surface wave is studied. The Floquet transform method is used to decompose Rayleigh surface wave into a set of spatial harmonic waves. Considering the periodic condition of the viaduct, the wave number domain dynamic response of the periodic viaduct on saturated soil subjected to Rayleigh surface wave excitation is obtained by the transfer matrix method. Then the space domain dynamic response is retrieved by means of the inverse Floquet transform. Numerical results show that when the periodic viaduct is undergoing in-plane vibration, there exist three kinds of characteristic waves corresponding to axial compression, transverse shear, and bending vibration. Furthermore, when the frequency of Rayleigh wave is within the pass band of the periodic viaduct, the disturbance propagates over a very long distance and the attenuation of the wave motion far from the source is determined by the characteristic wave with the smallest attenuation, while the vibration attenuates rapidly and propagates in a short distance when the frequency of excitation source is in the range of band gap of periodic structure.

#### 1. Introduction

As we know, viaduct structure is widely used in engineering. Usually, the viaduct structure generally has equal spans, which means the distance between adjacent piers of a multispan viaduct structure is constant; it can be considered as periodic structure, with the basic element consisting of three parts: a pier, two longitudinal beams, and three linking springs. The period is the distance between two neighboring piers.

The periodic structure has a significant vibration characteristic where energy band exists in periodic structure [1, 2]. When elastic wave propagates in a periodic structure, the vibration within a certain frequency range cannot be passed, which is called band gap, and the vibration within a certain frequency range can be passed, which is called pass band. It provides a new idea for the seismic design and vibration control by using vibration characteristics of periodic structures. With the rational design of the periodic viaduct in geometry and material parameters, it can ensure the frequency of main seismic waves is in the band gap of the viaduct structure, which effectively reduces the structure vibration and damage caused by earthquake wave. Otherwise, if great energy seismic waves are difficult to pass the viaduct structure, sharp increase of energy may be caused in the structural. Therefore, considering the energy band principle of periodic structures, the seismic design and vibration control measures can be achieved by adjusting the structure itself without additional structures.

At present, there are many seismic design methods about viaduct, such as Response Spectrum Method [3–5], Time-History Analysis Method [6–8], and Random Vibration Method [9–11]. In the above methods, the viaduct is generally simplified as single degree of freedom system or multidegree of freedom system, and seismic wave is simulated by standing wave. Clearly, the analysis using standing wave method cannot reflect the propagation characteristics of vibration wave in the periodic viaduct. As for the structural vibration induced by seismic wave, the viaduct piers attached to soft foundation were firstly excited. So the seismic energy passed on the viaduct structure varies with the distance from the viaduct pier to vibration source location, usually being nonlinear spatial distribution [12, 13]. Therefore, it is necessary to establish mathematical model which can reflect the propagation characteristics of nonuniform seismic wave in the periodic viaduct so as to provide theoretical basis for seismic design of the viaduct structure.

The periodic viaduct model, consisting of one pier, two longitudinal beams, and three springs, is established to describe the multispan viaduct structure [14, 15]. In this study, as for the Rayleigh surface wave at the bottoms of the piers on the saturated soil, the Floquet transform method is introduced to decompose it into a set of spatial harmonic waves. Considering the periodic condition of the viaduct, the eigenequation for the in-plane vibration of the viaduct in the wave number domain is obtained through transfer matrix method. Then the response in spatial domain of the periodic viaduct on saturated soil under Rayleigh waves can be retrieved by means of the inverse Floquet transform. The influence caused by the characteristic wave propagating in periodic structures and the different Rayleigh waves is discussed.

#### 2. Control Equations of Periodic Viaduct In-Plane Vibration

Figure 1 shows the periodic viaduct structure with infinite number of spans. The track, track plates, and beams of each span are simplified as left and right horizontal beams, which are connected by a pier supported on a semi-infinite saturated ground. The intermediate track between neighboring ballasts connecting the left and right horizontal beams is simulated by spring, which is assumed to be able to support axial force, shear force, and bending moment; and the jointing elements between piers and the left or right horizontal beam are also simulated by springs. Therefore, each unit of the periodic viaduct includes a pier, two horizontal beams, and three springs.