Shock and Vibration

Volume 2016 (2016), Article ID 1928086, 14 pages

http://dx.doi.org/10.1155/2016/1928086

## Vortex-Induced Vibration of a Cable-Stayed Bridge

^{1}Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang, Jiangsu 212013, China^{2}School of Astronautics, Harbin Institute of Technology, P.O. Box 137, Harbin 150001, China^{3}Department of Mechanical Engineering, University of Maryland, Baltimore County, Baltimore, MD 21250, USA

Received 21 June 2015; Revised 13 October 2015; Accepted 15 October 2015

Academic Editor: Tai Thai

Copyright © 2016 M. T. Song et al. 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 response of a cable-stayed bridge that consists of a simply supported four-cable-stayed deck beam and two rigid towers, subjected to a distributed vortex shedding force on the deck beam with a uniform rectangular cross section, is studied in this work. The cable-stayed bridge is modeled as a continuous system, and the distributed vortex shedding force on the deck beam is modeled using Ehsan-Scanlan’s model. Orthogonality conditions of exact mode shapes of the linearized undamped cable-stayed bridge model are employed to convert coupled governing partial differential equations of the original cable-stayed bridge model with damping to a set of ordinary differential equations by using Galerkin method. The dynamic response of the cable-stayed bridge is calculated using Runge-Kutta-Felhberg method in MATLAB for two cases with and without geometric nonlinear terms. Convergence of the dynamic response from Galerkin method is investigated. Numerical results show that the geometric nonlinearities of stay cables have significant influence on vortex-induced vibration of the cable-stayed bridge. There are different limit cycles in the case of neglecting the geometric nonlinear terms, and there are only one limit cycle and chaotic responses in the case of considering the geometric nonlinear terms.

#### 1. Introduction

Vortex-induced vibration (VIV) of a long-span structure is of practical importance to bridge engineering after collapse of the Tacoma Narrows bridge in 1940 [1]. VIV of a structure immersed in a fluid flow results from forces generated by alternating shedding of vortices from its surface. The structural vibration interacts with the flow, changing the fluid forces acting on the structure, and strongly nonlinear structural response with multifrequencies takes place [2]. VIV may lead to failure of a cable-stayed bridge due to fatigue damage and affect travel safety and/or comfort levels of its occupants [3]. Hence, an accurate prediction of the response of the cable-stayed bridge to vortex shedding at an early design stage is essential.

To achieve this objective, computational fluid dynamics (CFD) techniques are widely adopted to compute fluid forces on the structure by calculating the flow field information. Major CFD approaches, including direct numerical simulation [4–7], the time-marching scheme [8], and the vortex-in-cell method [9–12], mostly directly or approximately solve the time-dependent Navier-Stokes equation; however, they are limited by heavy computational requirement, which is difficult to satisfy up to now.

Apart from numerical simulations, semiempirical models have emerged as an alternative approach for predicting VIV due to their simple forms. A detailed review on VIV modeling has been given by Gabbai and Benaroya [13], according to which semiempirical models can be divided into two main classes: single-degree-of-freedom (SDOF) models and wake-oscillator models. The former can be classified into negative-damping models [14–17] and force-coefficient data models [18–20]. The wake-oscillator models consider two variables: a structural response variable and a fluid dynamic variable (e.g., the lifting force) [21–26].

The above semiempirical models are not able to predict the structural response for any cross section shape of a bluff body since their model parameters rely on values of structural mass and damping. An empirical model of VIV of line-like structures with complex cross sections such as bridge decks, which requires few and relatively simple wind-tunnel tests, may be useful in practical applications. Ehsan and Scanlan [27] proposed a SDOF model referred to as Ehsan-Scanlan’s model, which satisfies the above requirement; a single wind-tunnel test with a relatively simple experimental setup, called the decay-to-resonance test, is needed to estimate its model parameters. Moreover, aeroelastic parameters identified on a section model can be used to calculate the response of a cable-stayed bridge considering actual model properties of the structure. Marra et al. [28] applied Ehsan-Scanlan’s model in a realistic case study and proposed an alternative identification procedure based on direct numerical solution of a nonlinear ordinary differential equation.

Most previous studies mainly focus on VIV of cylindrical bodies and a deck-shaped body to study VIV of stay cables and a deck beam, respectively, which are two main components of a cable-stayed bridge. However, there is interaction between the stay cables and deck beam when they vibrate [29]. This paper presents VIV of a cable-stayed bridge that consists of a simply supported four-cable-stayed deck beam and two rigid towers, and it aims to study effects of the geometric nonlinearities of stay cables on the deck beam with a uniform rectangular cross section that is subjected to a vortex shedding force. Nonlinear and linear partial differential equations that govern transverse and longitudinal vibrations of the stay cables and transverse vibrations of segments of the deck beam, respectively, were derived along with their boundary and matching conditions using a Newtonian approach. Ehsan-Scanlan’s model is used to model the vortex shedding force that is considered as a distributed force. Exact natural frequencies and mode shapes of the linearized undamped cable-stayed bridge model obtained in [29] are used to spatially discretize coupled governing partial differential equations of the original nonlinear cable-stayed bridge model with damping via Galerkin method. The dynamic response of the cable-stayed bridge is obtained by solving resulting nonlinear ordinary differential equations using Runge-Kutta-Felhberg method. Convergence of Galerkin method for VIV of the cable-stayed bridge is investigated. Numerical results show that there are significant influences of the stay cables on VIV of the deck beam: there are different limit cycles when one neglects geometric nonlinear terms associated with the stay cables, and there are only one limit cycle and chaotic responses in the case when the geometric nonlinear terms are considered.

#### 2. Problem Formulation

Consider a cable-stayed bridge that consists of a simply supported four-cable-stayed deck beam and two towers, subjected to vortex shedding on the deck beam, as shown in Figure 1. The deck beam consists of seven segments separated by its junctions with the stay cables and towers. The following assumptions are made in this work in the formulation of the vibration problem of the cable-stayed bridge model subjected to vortex shedding:(1)The cable-stayed bridge is modeled as a planar system.(2)The towers, to which the stay cables are attached, are built on a hard rock foundation and can be assumed to be rigid [29, 30]; they are connected to the deck beam through roller supports.(3)The stay cables and deck beam have linear elastic behaviors.(4)Each segment of the deck beam obeys the Euler-Bernoulli beam theory.