The dynamic response of a cablestayed bridge that consists of a simply supported fourcablestayed 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 cablestayed bridge is modeled as a continuous system, and the distributed vortex shedding force on the deck beam is modeled using EhsanScanlan’s model. Orthogonality conditions of exact mode shapes of the linearized undamped cablestayed bridge model are employed to convert coupled governing partial differential equations of the original cablestayed bridge model with damping to a set of ordinary differential equations by using Galerkin method. The dynamic response of the cablestayed bridge is calculated using RungeKuttaFelhberg 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 vortexinduced vibration of the cablestayed 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.
Vortexinduced vibration (VIV) of a longspan structure is of practical importance to bridge engineering after collapse of the Tacoma Narrows bridge in 1940 [
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 [
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 [
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 linelike structures with complex cross sections such as bridge decks, which requires few and relatively simple windtunnel tests, may be useful in practical applications. Ehsan and Scanlan [
Most previous studies mainly focus on VIV of cylindrical bodies and a deckshaped body to study VIV of stay cables and a deck beam, respectively, which are two main components of a cablestayed bridge. However, there is interaction between the stay cables and deck beam when they vibrate [
Consider a cablestayed bridge that consists of a simply supported fourcablestayed deck beam and two towers, subjected to vortex shedding on the deck beam, as shown in Figure
The cablestayed bridge is modeled as a planar system.
The towers, to which the stay cables are attached, are built on a hard rock foundation and can be assumed to be rigid [
The stay cables and deck beam have linear elastic behaviors.
Each segment of the deck beam obeys the EulerBernoulli beam theory.
Schematic of a cablestayed bridge that consists of a simply supported fourcablestayed deck beam and two rigid towers.
A free vibration analysis of the planar motion of this kind of cablestayed bridges without considering vortex shedding was presented in [
Let
The following nondimensional variables are introduced:
The Newtonian method is used here to derive nonlinear equations of motion of the cablestayed bridge model and a full set of geometric and dynamic boundary and matching conditions. Assuming that cable longitudinal inertial forces
The distributed vortex shedding force on the deck beam can be modeled using EhsanScanlan’s model [
Galerkin method is used to analyze the vibration of the cablestayed bridge. The dynamic response of stay cables and segments of the deck beam are expressed by
It should be noted that nonlinear terms
Geometric and physical parameters of a cablestayed bridge and aeroelastic parameters are listed in Table
Geometric and physical parameters of the cablestayed bridge and aeroelastic parameters.
Parameter  Unit  Value  

Deck beam  Mass per unit length of the deck beam ( 
kg/m  16940 
Elastic modulus of the deck beam ( 
N/m^{2}  2.0 × 10^{11}  
Area moment of inertia of the deck beam ( 
m^{4}  1.20  
Length of segment 
m  35  
Length of segment 
m  40  
Length of segment 
m  50  
Length of segment 
m  50  
Length of segment 
m  50  
Length of segment 
m  40  
Length of segment 
m  35  


Stay cables  Mass per unit length of the stay cables ( 
kg/m  286 
Elastic modulus of the stay cables ( 
N/m^{2}  2.0 × 10^{11}  
Crosssectional area of the stay cables ( 
m^{2}  0.0362  
Length of stay cable 
m  52  
Length of stay cable 
m  60  
Length of stay cable 
m  60  
Length of stay cable 
m  52  
Sagtospan ratios of the stay cables ( 
0.01  


Aeroelastic parameters  Air density ( 
kg/m^{3}  1.205 

1122.5 [  

6.88 [ 
Convergence of Galerkin method for given initial conditions
Effects of the number of Galerkin truncation terms on transverse displacements of the midpoint of the deck beam when there are no geometric nonlinear terms.
Phase portraits of the response of the midpoint of the deck beam for different numbers of Galerkin truncation terms,
Time response of the midpoint of the deck beam,
Time response of the midpoint of the deck beam,
Solutions of a reducedorder model for a flow dynamic system can converge to a spurious limit cycle after longtime integration, even if it is initialized with a correct configuration [
In many cases, higher mode shapes of the cablestayed bridge would be excited; one such case is that when there are vehicles moving on the deck beam of the bridge. Hence, the initial dynamic configuration of the cablestayed bridge can correspond to its higher mode shapes in the dynamic analysis of the cablestayed bridge subjected to a distributed vortex shedding force. It is obvious that Galerkin truncation with one term is not enough in these cases. Through the same method with
Time response of the midpoint of the deck beam,
Two different stable limit cycles corresponding to different initial configurations.
Twodimensional projections of phase portraits onto the
Twodimensional projections of phase portraits onto the
Convergence of Galerkin method for given initial conditions
Effects of the number of Galerkin truncation terms on transverse displacements of the midpoint of the deck beam when there are geometric nonlinear terms.
Phase portraits of the response of the midpoint of the deck beam for different numbers of Galerkin truncation terms,
Twodimensional projections of phase portraits onto the
Time response of the midpoint of the deck beam;
Bifurcation diagram of the cablestayed bridge with respect to
Chaotic response of the midpoint of the deck beam,
Time response of the midpoint of the deck beam,
Time response of the midpoint of the deck beam,
Phase portraits of the response of the midpoint of the deck beam for different initial conditions,
Chaotic response of the midpoint of the deck beam,
The dynamic behavior of a cablestayed bridge that consists of a simply supported fourcablestayed deck beam and two rigid towers subjected to a distributed vortex shedding force on the deck beam has been investigated. The dynamic response of the cablestayed bridge is calculated using Galerkin method in conjunction with RungeKuttaFelhberg method in MATLAB. Convergence of Galerkin method for the dynamic response of the cablestayed bridge is studied. Numerical simulations show that the geometric nonlinearities of the stay cables have significant influence on VIV of the cablestayed bridge, and further conclusions can be summarized as follows:
In the case when the geometric nonlinear terms are neglected, accurate calculation of the response amplitude of the cablestayed bridge at lockin only needs use of the first mode shape of the linearized undamped cablestayed bridge model when the initial dynamic configuration of the cablestayed bridge corresponds to its mode shape whose mode number is smaller than seven. There is a different limit cycle when the initial dynamic configuration corresponds to its mode shape whose mode number is equal to or larger than 7.
In the case when the geometric nonlinear terms are considered, calculation of the response of the cablestayed bridge generally needs use of multiple mode shapes of the linearized undamped cablestayed bridge model even when the initial dynamic configuration of the cablestayed bridge corresponds to its first mode shape. There is a limit cycle when the initial dynamic configuration of the cablestayed bridge corresponds to its first mode shape and its amplitude is smaller than 0.3 for the generalized coordinate, and there is chaotic response when the initial dynamic configuration of the cablestayed bridge corresponds to its first mode shape with its amplitude larger than 0.3 for the generalized coordinate or one of its higher mode shapes.
The authors declare that there is no conflict of interests regarding publication of this paper.
This work is supported by the National Natural Science Foundation of China under Grant nos. 11302087 and 11442006, the Natural Science Foundation of Jiangsu Province under Grant no. BK20130479, the Research Foundation for Advanced Talents of Jiangsu University under Grant no. 13JDG068, and the National Science Foundation under Grant no. CMMI1000830.