Mathematical Problems in Engineering

Volume 2016, Article ID 3560160, 12 pages

http://dx.doi.org/10.1155/2016/3560160

## Dynamic Behavior of a Prestressed Concrete Bridge with a Switching Crack Subjected to Moving Trains

College of Civil and Transportation Engineering, Hohai University, Nanjing, China

Received 28 June 2016; Accepted 25 July 2016

Academic Editor: Yan-Wu Wang

Copyright © 2016 Chunyu 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

Concrete bridges chronically exposed to the natural environment are vulnerable to cracking. Under prestressing forces, the crack may close, but it will open under large dynamic loads, such as heavy trains. So the crack state can switch during the vibration. This paper investigates the nonlinear dynamic behavior of the prestressed concrete bridge with such a switching crack subjected to moving trains. Firstly, the finite element method is adopted to formulate the motion equations of train-bridge system while the crack state remains constant. Then the displacement components are analyzed to investigate the effect of static loads at the switching instant. Finally, an iterative algorithm is adopted to calculate the nonlinear responses. The proposed method is verified by the calculation of an actual prestressed concrete bridge, and the results show that the crack switching can increase the peak of the bridge displacement during the vibration, cause an obvious nonlinearity between the displacement and the vehicle mass, and make the displacement become more variable with the train velocity.

#### 1. Introduction

Prestressed concrete bridges have found wide applications in railway engineering in recent years. Because of being chronically exposed to the natural environment, they are vulnerable to cracking under heavy trains, seismic excitation, and other loads. When the bridges are subjected to the independent action of static loads, especially prestressing forces, the cracks may be closed. However, if large dynamic loads, such as heavy trains, are present, the cracks will open and close in time depending on the structural vibration amplitude [1]. Various studies over the last decade have shown that a structure with such cracks exhibits nonlinear dynamic behavior [2], and its safety and serviceability are seriously affected. So it is essential to study the vibration of the prestressed concrete bridge with such cracks under moving trains.

In recent years, the vibration of cracked structures subjected to dynamic loads has attracted more and more attentions of researchers, and many methods for analyzing the vibration were proposed. Generally, these methods can be grossly divided into two kinds according to crack models used.

The first kind is the modal method for the structure with switching cracks, which are either fully open or fully closed and can switch their state instantaneously, showing a bilinear behavior. The switching condition is assumed to be determined by the sign of the normal strain [3], the displacement [4], or the curvature [5] near the crack tip. When the crack is fully open, it may seriously affect the local stiffness and response of the beam. Its effect can be modeled by means of a rotational spring model [6] or a crack disturbance function [7, 8]. The overall behavior of the structure can be considered as a sequence of linear states, each of which can be evaluated through a modal analysis [5].

Through the modal method, Law and Zhu [9] built an interaction model between the cracked beam and a moving vehicle, which was validated by an experimental test performed on a reinforced concrete beam with a T-section. Jaksic et al. [10], respectively, calculated the eigenvalues of open and closed crack states to formulate the vibration of cracked bridges under a moving oscillator and concluded that the statistical properties of the bridge response were sensitive to the presence of switching crack. Fu [11] investigated the effect of switching cracks on the vibration of a continuous beam bridge subjected to moving vehicles and found that, compared to open cracks, switching cracks could result in higher acceleration and increase the high modal contribution to the displacement.

The second kind of method is mainly concerned with the structure with breathing cracks, for which there is a smooth transition phase between open and closed crack states. So the cracks can open and close continuously, and the crack state is assumed to be dependent on the responses, such as the contact condition at the crack interfaces [12], the time [13], or the curvature [1]. As the structural stiffness can vary with the crack state, it will change gradually during the vibration, and the structural behavior will exhibit obvious nonlinearity.

Ariaei et al. [1] directly used a discrete element technique to calculate the vibration of a beam with open and breathing cracks subjected to a moving mass and concluded that a beam with a breathing crack had less deflection compared to a beam with an open crack. Nguyen [14] adopted the finite element method to analyze the structural stiffness change before and after the breathing and used the instantaneous frequency of the responses under a moving vehicle to discriminate between open and breathing cracks. Andreaus et al. [2] used two-dimensional finite elements to solve the vibration of a cantilever beam with a breathing crack under the harmonic loads and impulsive loads and simulated the behavior of the crack as a frictionless contact problem.

From the aforementioned researches, it is seen that all these methods could analyze the structures with switching or breathing cracks and take the effect of dynamic loads into account. But there are few methods considering the effect of static loads. Most of researches about the vibration of prestressed concrete beams focused on the effect of prestressing force on the modal parameters. For example, Saiidi et al. [15], Kanaka and Venkateswara [16], and Miyamoto et al. [17] thought that the presence of prestressing forces decreases the natural frequencies due to the “compression softening” effect. But Hamed and Frostig [18] and Jaiswal [19] held the view that prestressing has no or negligible effect on the natural frequencies of prestressed beams.

For the prestressed concrete bridge with a switching crack, static loads including prestressing forces and the self-weight loading still persist during the vibration and may change the bridge time-history responses. It is because the sum of static and dynamic responses decides the state of the switching crack which finally affects the structural stiffness. Therefore, in this study, the dynamic behavior of the prestressed concrete bridge with a switching crack will be studied under a moving train. Firstly, motions equations of train-bridge system are firstly formulated through the finite element method; then the response components are analyzed at the switching instant to investigate the effects of static loads, and an iterative algorithm is adopted to obtain the bridge responses.

#### 2. Vibration of a Prestressed Concrete Bridge with a Switching Crack

##### 2.1. Model of Train-Bridge System

As shown in Figure 1, a prestressed concrete bridge with simple supports is considered. The axial direction of the bridge is taken as the -axis and the vertical direction as the -axis. It is assumed that the bridge mass and cross section are uniform along the -axis, and all deformations are small enough that an orthogonal coordinate system can be used.