Proceedings of the 2018 World Transport Convention Beijing, China, June 18-21, 2018

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Numerical Simulation and Aerodynamic Energy Analysis of Limit

Cycle Flutter of a Bridge Deck

Xuyong Ying State Key Laboratory of Safety and Health for In-Service Long Span Bridges,

Jiangsu Transportation Institute Co. Ltd. No. 2200 Chengxin Ave., Nanjing, P.R. China

yingxuyong@hotmail.com

Zhen Sun State Key Laboratory of Safety and Health for In-Service Long Span Bridges,

Jiangsu Transportation Institute Co. Ltd. No. 2200 Chengxin Ave., Nanjing, P.R. China

sz21@jsti.com

Fuyou Xu School of Civil Engineering, Dalian University of Technology

No.2 Linggong Road, Dalian, P.R. China fuyouxu@hotmail.com

ABSTRACT

Based on ANSYS FLUENT, a fluid-structure interaction (FSI) model has been developed to investigate the limit cycle flutter (LCF) characteristics of a bridge deck. Its accuracy is verified by simulating the critical flutter state of one thin plate section with theoretical solutions. The unsteady flows around an elastically supported deck at different wind speeds and structural damping ratios are simulated. The LCF phenomenon of the bridge deck is well captured by the present numerical model. The simulated results indicate that the LCF amplitude increases gradually with the wind speed until the divergent flutter occurs. The structural damping has remarkable influence on the critical wind speed and the LCF amplitude. The characteristics of aerodynamic energy input and dissipation during LCF are detailedly investigated. The total mechanical energy varies bounded during LCF, and it remains unchanged in a vibration period. For the LCF occurrence, the work done by aerodynamic forces and the dissipated energy by structural damping should cancel out. The developed numerical simulation approach can help to serve as a building block for developing an overall analysis framework for investigating the LCF characteristics of long-span bridges.

KEYWORDS: Limit cycle flutter; CFD; Bridge deck; Aerodynamic energy; Unsteady flow.

1 INTRODUCTION

Flutter analysis is classically based on Scanlan's linear self-excited forces model (Scanlan and Tomko, 1971), in which the bridge deck undergoes harmonic vibration with net zero damping at critical flutter state. According to the Scanlan's model, the self-excited forces are the linear functions of deck motions and flutter derivatives. Actually, the self-excited forces inevitably contain nonlinear components due to the bluff configurations and large amplitudes, and therefore, nonlinear aeroelastic phenomenon may be observed for some cases of bridge flutter. For numerous wind tunnel tests on bluff and/or streamlined deck sections with large angle of attack, when the wind speed is sufficiently high, the decks experience limit cycle oscillation (LCO), rather than the divergent flutter (Xu and Chen, 2008; Long, 2010; Zhu and Gao, 2015). This kind of LCO can be classified as the limit cycle flutter (LCF), in which the deck reaches steady-state vibration for a specific wind speed. Zhang

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(2007) presented a single-degree-of-freedom (SDOF) nonlinear aerodynamic model, by which the phenomenon of LCF was well explained. Zhu and Gao (2015) carried out a series of wind tunnel tests on several typical bridge deck section models to investigate their characteristics of LCF. The results showed that the LCF is coupled vertical and torsional vibration with a single frequency.

The usefulness of energy concept in investigating the flutter phenomenon is well known. In recent years many researches on explaining the flutter mechanism have been conducted from the energy standpoint. In order to gain insight into the understanding of multimode flutter and its generation mechanism, the aerodynamic coupling among modes and its damping or exciting roles are investigated from the viewpoint of system energy by Chen et al. (2000). They concluded that the dominant modes for most coupled flutter are the fundamental symmetric vertical and torsional modes. The coupled self-excited forces acting on the central span are the main source of negative damping that leads to flutter. Bendiksen (2004) presented an analysis of the LCO of aircraft wings in order to investigate the mechanisms of energy transfer from the fluid to the structure. He pointed out that energy is extracted from the airstream at a sufficient rate to produce rapidly increasing amplitudes in the subsonic flutter. The LCO was the rule rather than the divergent flutter in the transonic region, because of the strong aerodynamic nonlinearities induced by moving shocks on the wing surface. Furthermore, an aeroelastic mode energy diagram for bending-torsion flutter was presented for explaining the occurrence of LCO in the transonic region. Liu et al. (2015) numerically simulated the flutter of a box girder by a fluid-structure calculation strategy and dynamic mesh technique. Results indicated that the windward nozzle of the box girder was the main pneumatic energy absorption area. The flutter stability criterion was also provided from the viewpoint of energy balance. Based on energy analyses, a large amount of researches have been performed to investigate the flutter mechanisms and control of bridge decks, e.g. Körlin and Starossek (2004), Hua and Chen (2008), Zhang et al. (2011), and Li et al. (2015). In recent years, the performances of the divergent flutter for bridge decks have been well studied. However, the LCF phenomenon has received less attention. The intensive investigations on the LCF are of great significance to construct more accurate bridge flutter theory and judging criterion for flutter instability.

In present paper, numerical simulation and aerodynamic energy analysis of LCF of a bridge deck are conducted using CFD techniques. Section 2 presents the fluid-structure interaction (FSI) model and solving strategy. Section 3 describes the structural parameter and geometry of the deck section used in present study. Section 4 focuses on the verification of the present numerical model by simulating the critical flutter state of a thin plate section. Section 5 analyzes the LCF response of the bridge deck for various wind speeds. The characteristics of aerodynamic energy input and dissipation during LCF are investigated in Section 6.

2 NUMERICAL MODEL DESCRIPTIO

2.1 Governing Equations for Fluid

The incompressible, unsteady, 2-D air flow with moving boundaries can be modeled by means of the Reynolds Averaged Navier-Stokes (RANS) equations. For the numerical aeroelastic simulation that contains dynamic meshes, an important requirement is the accurate simulation for the interactions between air flow and moving deck section. In the present study, the governing equations are given in Arbitrary Lagrange-Euler (ALE) formulation, which accommodates the moving boundaries and any subsequent deformation of the underlying discrete mesh. By introducing the grid velocity mju of the moving mesh, the ALE formulation for mass and momentum of incompressible fluid may be expressed as

( )0j mj

j

u ut x

ρρ ∂ −∂+ =

∂ ∂ (1a)

( )( )j mj ii i

eff ij i j j

u u uu up St x x x x

ρρµ

∂ −∂ ∂∂ ∂+ = − + +

∂ ∂ ∂ ∂ ∂ (1b)

where iu or ju , ρ , jx and p are the fluid velocity components, the fluid density, the Cartesian

spatial coordinates and the fluid pressure, respectively; iS denotes the additional momentum source

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contributions, if any; effµ is the effective viscosity which includes laminar and turbulent contributions (Hassan et al., 2010). In the RANS approach, the turbulence viscosity is modeled by the SST k ω− model (Menter, 1994). Ying et al. (2012) conducted a comprehensive simulation of unsteady flow around rectangular cylinders by using different RANS approaches, and the SST ω−k model is found to be the best choice among various RANS models and has accuracy enough to be suitable for practical problems. Further details on the SST turbulence model implementation may be referred to the reference by Menter (1994).

The ANSYS FLUENT adopted in this study uses the finite-volume method (FVM) to solve the fluid governing equations. The ALE formulation of governing equation enables the conservative fluid calculations with mesh adaptation in time. The discretization method of governing RANS equations remains unchanged from the general application of FVM (Schneider and Raw, 1987). The second order implicit scheme and upwind scheme are used for the time and spatial discretization, respectively. SIMPLE (semi-implicit pressure linked equations) algorithm is used for solving the discretized governing equations. The time-step is set as Δt=0.001s. 2.2 Computational Domain and Mesh Arrangement

The computational domain and boundary conditions are schematically shown in Figure 1 for a two dimensional x-y slice. Due to large vibration amplitudes may be concerned, the wide rectangular computational domain is set as 16.0×5.0 m. The boundaries are sufficiently far away from the sections so as to eliminate the flow obstacle effect on the inflow and outflow boundary conditions. At the inflow boundary, the flow with a low turbulence intensity of 0.5% is used. A non-slip condition is used for the section surfaces and the top and bottom surface of the domain. It is assumed that the flow is fully developed at the outlet boundary.

Figure 1: Computational domain and boundary conditions (unit: m).

Considering the mesh number should be as low as possible to obtain high efficiency in

computation, the hybrid grids are used for the whole computational domain. The structured quadrangular grid is generated for the zone in the vicinity of deck section, while the unstructured triangular grid is adopted for the outside zone. The height of 0.0001 m is selected for the first layer close to deck surface and the total number of mesh cells is 86136, which is adequate to resolve the velocity and viscous sublayer. A view of meshes near the surface of the deck section is depicted in Figure 2.

Figure 2: Mesh near the surface the deck section.

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For fluid-structure interaction (FSI) problems, the section model is subjected to heaving and/or torsional vibration, and the dynamic mesh technique is employed to fulfill this target. The present whole computational domain is decomposed into two sub-domains those similar to the early work by Fransos and Bruno (2006), as shown in Figure 2. The inner oval-shaped region is rigidly connected with the sectional model and move synchronously, while the mesh in outer region deforms at each iteration time-step. The spring-based smoothing technique is adopted to accommodate the deformation of dynamic mesh. Not any deformation occurs for the rigid mesh region and their quality remains unchanged throughout the whole calculation process, which is a benefit especially for solving the viscous sub-layer that close to the section surface. The more detailed descriptions for dynamic mesh technique may be found in (Xu et al., 2014).

2.3 Governing Equations for Deck Motion

For a 2-D rigid section, its planar vibrations can be described in terms of three displacement components, i.e., p, h and α, defined at the shear center. p and h are the translational motion displacement in the x and y directions as shown in Figure 3, respectively, and α is the torsional displacement about the shear center. The aeroelastic drag force, lift force, and twist moment induced by lateral motion have been generally regarded as insignificant to bridge flutter. Thus, a typical 2-D section model with 2-DOF, i.e., h and α, is used in the formulation of present aeroelastic simulation. The torsional displacement α is taken positive clockwise.

Figure 3: Structural reference system.

The deck model is assigned mass (m, I), stiffness (kh, kα) and viscous damping coefficients (ch,

cα), and the motion equations for the section model can be expressed as ( ) ( ) ( ) ( )h h Lmh t c h t k h t F t+ + =&& & (2a)

( ) ( ) ( ) ( )TI t c t k t M tα αα α α+ + =&& & (2b) where ( )h t&& , ( )h t& , ( )h t are the vertical acceleration, velocity, and displacement, respectively; ( )tα&& ,

( )tα& , ( )tα are the torsional acceleration, velocity, and displacement, respectively; ( )LF t and ( )TM t are the aerodynamic lift and twist moment acting on the section model, respectively. The fourth-order Runge-Kutta method is used to discretely solve the structural motion Eq. (2). For the vertical motion, the vertical motion Eq. (2a) can be expressed as a first order differential equation:

2( , ) ( ) / 2 ( ) ( )L h h hf h h F t m h t h tξ ω ω= − −& & (3) where

2hh h

k fm

ω π= = (4)

2h

hh

ck m

ξ = (5)

where hω , hf , and hξ denote the natural circular frequency, the natural frequency, and the damping

ratio of vertical bending, respectively. The velocity ( )h t& and displacement ( )h t can be obtained by discretizing Eq. (2a):

( )1 2 3 4( ) ( ) 2 26th t t h t K K K K∆

+ ∆ = + × + + +& & (6)

5

( ) ( )2

1 2 3( ) ( ) ( )6t

h t t h t t h t K K K∆

+ ∆ = + ∆ × + × + +& (7)

where 2

1( ) 2 ( ) ( )L

h h hF tK h t h t

mξ ω ω= − −& (8a)

22 1

( ) 2 ( ) ( ) ( )2 2

Lh h h

F t t tK h t K h t h tm

ξ ω ω∆ ∆ = − + − +

& & (8b)

( )22

3 2 1( ) 2 ( ) ( ) ( )

2 2 4L

h h h

tF t t tK h t K h t h t Km

ξ ω ω ∆∆ ∆ = − + − + +

& & (8c)

( )22

4 3 2( ) 2 ( ) ( ) ( )

2 2L

h h h

tF t tK h t K h t t h t Km

ξ ω ω ∆∆ = − + − + ∆ ⋅ +

& & (8d)

The discretized Eq. (6)~(7) can be solved with the initial conditions of ( 0) 0h t = =& and ( 0) 0h t = = . The torsional motion Eq. (2b) for the section model can also be solved by the same

method, which is omitted herein for brevity. 2.4 The Fluid-structure Coupling Strategy

In the FSI analysis for aeroelastic problems, it is significant to satisfy the geometrical compatibility and the equilibrium conditions on the interface between the fluid and the structure (Zhang and Hisada, 2004). For this purpose, the strong coupling method or the weak coupling method has often been used to solve the FSI system. For the strong coupling method, sometimes referred to as directly coupling or simultaneous solution by some researchers, the variables of the coupled system is solved and corrected simultaneously. However, the use of strong coupling method is limited due to the less flexibility between the fluid and structure solver in the time integration. It also needs to modify the existing fluid dynamic solver and structure dynamic solver. Instead of directly solving the coupling system of Eq. (1) and (2), a weak coupling method is used to solve this fluid-structure coupling problem, by which the fluid and structural governing equations are sequentially solved and the interface conditions are satisfied in an iterative manner. Therefore, the existing solvers for fluid and structure can be easily adopted, and the flexibility between the fluid and structure solver is maintained. Furthermore, the weak coupling method requires less storage and computational time compared with the strong coupling method.

The flow chart of the computational process is shown in Figure 4. Initially, the fully developed flow around the stationary deck section can be obtained. Meanwhile, the current flow condition provides the basis for the subsequent fluid-structure coupling simulation. At that moment, the corresponding lift force and torsional moment may be calculated by integrating the pressures and shear forces along the section surfaces. Then the calculated aerodynamic forces are employed as an input into the structural motion solver to predict the displacement and velocity of the deck section. This velocity is then used to determine the new position at the next time-step. The ANSYS FLUENT re-meshes the computational domain in response to new geometry configuration. A series of aeroelastic problems can be efficiently simulated by this fluid-structure coupling strategy.

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Figure 4: Flow chart of the computational process.

3 DESCRIPTION OF DECK SECTION AND SIMULATION CASES

Two representative sections are included in this study, and their scaled sections are depicted in

Figure 5. The first section is a thin flat plate, whose theoretical value of critical flutter speed is available, by which the validity and accuracy of the present numerical model can be verified. The second section originated from a streamlined cross-section of a steel box girder of the Taohuayu self-anchored suspension bridge (main span length: 406 m) in China. The aeroelastic tests for this deck section were conducted using the spring-supported section model in the wind tunnel laboratory by Long (2010). Table 1 shows the parameters of the section model in the wind tunnel tests. As shown in Long (2010), the LCF phenomenon was observed at the +5° initial attack angle for the section model without handrails.

At +5° initial attack angle with different wind speed U0, the wind-induced response of the deck model can be calculated by the numerical simulations, where the deck model is allowed to vibrate under the specified structural parameters as given in Table 1. The torsional and vertical responses (include displacement and velocity of the motion) are monitored to evaluate the vibration performances of the deck section.

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Figure 5: Sectional model of thin plate and Taohuayu Bridge deck (unit: mm).

Table 1: Parameters of the section model.

Parameters Symbols Thin Plate Bridge Deck Width of the section model B (m) 0.6 0.78 Height of the section model H (m) 0.002 0.07 Mass of the section model m (kg/m) 16.82 12.003 Mass moment of the section model I (kg.m2/m) 0.5856 0.4375 Natural frequency of vertical motion fh (Hz) 1.627 2.01 Natural frequency of torsional motion fα (Hz) 2.973 3.786 Damping ratio of vertical motion ξh 0.006 0.005 Damping ratio of torsional motion ξα 0.008 0.005

4 VERIFICATION OF THE PRESENT FSI MODEL

Figure 6 shows the calculated time history of dynamic response for the thin plate section. The dynamic response of the thin plate almost decay to zero after initial perturbation at U0 = 19.25 m/s, while the divergent vibration occurs at U0 = 19.30 m/s. It indicates that the critical flutter speed of the thin plate is between 19.25~19.30 m/s according to the simulated results. The theoretical analysis results from complex model method and the results from present simulations are compared in Table 2. As can be seen, the numerically simulated critical flutter speed and frequency are in good agreement with the theoretical solutions, which verifies the accuracy of the present numerical model.

0 2 4 6 8 10 12-0.06

-0.04

-0.02

0.00

0.02

0.04

Torsinal displacement α Vertical displacement h/B

Dyna

mic

resp

onse

Time (s)0 2 4 6 8 10 12

-0.08

-0.04

0.00

0.04

0.08

Torsinal displacement α Vertical displacement h/B

Dyna

mic

resp

onse

Time (s) (a) U0 = 19.25 m/s, f=2.145 Hz (b) U0 = 19.3 m/s, f=2.143 Hz

Figure 6: Time history of dynamic response, thin plate section.

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Table 2: Critical flutter wind speed and frequency of the thin plate.

Calculation method Critical flutter wind speed (m/s) Flutter frequency (Hz) Complex model method 19.35 2.165 Numerical simulation 19.25~19.30 2.143~2.145

5 ANALYSIS OF WIND-INDUCED RESPONSE

For the case of initial attack angle α0=+5°, the wind-induced response of deck section under different wind speeds can be calculated by conducting a series of numerical simulation. The torsional and vertical responses (include displacement and velocity of the motion) are monitored to evaluate the wind-induced vibration characteristics of the bridge deck. The time history of torsional and vertical displacements of the deck section for different wind speeds can be seen in Figure 7.

(1) At U0=9 m/s, the motion type is attenuated vibration with the dynamic response converging to equilibrium position after initial perturbation. The vertical motion have large aerodynamic damping when the wind speed approaches the critical flutter wind speed, results in the rapid attenuation of vertical displacement.

(2) When the wind speed reaches U0=12 m/s, the torsional and vertical displacements increase gradually with flow time in the beginning, and then the motion changes from divergent vibration to steady-state vibration. The maximum of the amplitude keep bounded both for torsional and vertical motion, which indicates that the LCF occurs at U0=12 m/s. The aerodynamic energy balance during LCF will be detailed discussed in the following section.

(3) For the wind speed U0=13.5 m/s, the wind-induced displacements rapidly increase with flow time, which shows the divergent flutter occurs. It also indicates that the deck motion continuously extracts energy from the flow field for this case.

0 2 4 6 8 10

-0.06

0.00

0.06

0.12

0.18

0.24

Deck

disp

lacem

ents

t (s)

α h/B WM

-0.6

0.0

0.6

1.2

1.8

2.4

Mec

hani

cal e

nerg

y (J)

0 2 4 6 8 10 12

-0.05

0.00

0.05

0.10

0.15

Deck

disp

lacem

ents

t (s)

α h/B WM

-0.5

0.0

0.5

1.0

1.5

Mec

hani

cal e

nerg

y (J)

(a) U0 = 9 m/s (b) U0 = 12 m/s

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.07

0.00

0.07

0.14

0.21

0.28

Deck

disp

lacem

ents

t (s)

α h/B WM

-0.6

0.0

0.6

1.2

1.8

2.4

Mec

hani

cal e

nerg

y (J)

(c) U0 = 13.5 m/s

Figure 7: Time history of dynamic response and mechanical energy, ξ0=0.005.

The structural damping is one key factor in determining the critical flutter speed of bridge decks (Chen and Kareem, 2004). With different structural damping ratios, the torsional and vertical displacements at different wind speeds are numerically calculated, and the corresponding steady

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amplitudes are shown in Figure 8. The critical flutter wind speed increases with the structural damping ratio, which indicates that the structural damping is very efficient in controlling the LCF of bridge decks. At the same wind speed, the steady amplitude can be reduced by increasing the structural damping ratio. For different structural damping ratios, the increasing rates of torsional amplitude with wind speed remain essentially the same. However, for the vertical amplitude, the increasing rate is lower for the case of lower structural damping ratio.

The numerically simulated critical wind speed of LCF is fairly close to that of the experimental result, both are around 11 m/s. However, at the same wind speed, the simulated torsional amplitudes are relatively larger than that of the experimental results. These discrepancies may be ascribed to the following causes: (a) a quasi 3D model was used in wind tunnel tests, and a 2D model was used in the numerical simulations. The spanwise correlation effects exist in the experiments, while it’s not considered in the numerical simulations; (b) the structural nonlinearities in stiffness and damping are unavoidable for the experimental model, which were verified in some studies (Zhu and Gao, 2015; Lee and LeBlanc, 1986); however, these nonlinearities can be perfectly excluded for the numerical model; (c) the model geometry and the wall conditions inevitably have slightly differences between experimental tests and numerical simulations.

8 9 10 11 12 13 14

0

2

4

6

8

10 Num., ξ0=0 Num.,ξ0=0.005 Num.,ξ0=0.01 Num.,ξ0=0.02 Exp., ξ0≈0.005

α 0 (d

eg.)

U0 (m/s)8 9 10 11 12 13 14

0.000

0.006

0.012

0.018

0.024 ξ0=0 ξ0=0.005 ξ0=0.01 ξ0=0.02

h 0 (m

)

U0 (m/s) Figure 8: Effect of structural damping on steady-state amplitude: (a) torsional; (a) vertical

6 AERODYNAMIC ENERGY ANALYSIS

Let AW denote the work done by aerodynamic forces over a specified time interval 1 2[ , ]t t , 2

1

( ) ( ) ( ) ( )t

A h T LtW W W M t t F t h t dtα α = + = + ∫ && (9)

where Wα , hW are the work done by aerodynamic moment and lift, respectively. The structural damping always does negative work during LCF. The energy dissipated by

structural damping can be expressed as 2

1

2 2( ) ( )t

D htW c t c h t dtαα = + ∫ && (10)

During vibration, the sum of the kinetic and potential energy of the deck section is called its total mechanical energy, and it can be expressed as

[ ] [ ] [ ] }{ 22 2 2( ) ( ) ( ) 0.5 ( ) ( ) ( ) ( )M K P hE t E t E t I t m h t k t k h tαα α = + = + + + && (11)

where ME , KE , PE are the total mechanical energy, kinetic and potential energy of the deck section, respectively. Figure 7 shows the time history of mechanical energy at different wind speed. At U0 = 9m/s, with the attenuation of deck motion, the initial mechanical energy is dissipated by the air flow effectively. For the case of U0 = 13m/s, the mechanical energy rapidly increase with flow time, and result in the occurrence of divergent flutter. When the LCF occurs at U0 = 12m/s, the mechanical energy of the deck section increases initially over a finite time interval, and then it tends to be steady.

The total mechanical energy of the deck section can also be written in Hamiltonian form as

[ ]1

( , , )s

M i i i ii

E q p L q q t=

= −∑ & (12)

10

where ( , , )i i K PL q q t E E= −& is the Lagrangian and i ip L q= ∂ ∂ is the generalized momentum corresponding to the generalized aerodynamic coordinate iq ; s is the degree of freedom. Since the unsteady aerodynamic forces are non-conservative, the generalized forces acting on the deck section include aerodynamic forces and damping forces, which can be expressed as

( )ii i

d L LQdt q q

∂ ∂= −

∂ ∂& (13)

Taking ME with respect to time,

( )1 1 1

( ) ( )s s s

Mi i i i i i A D

i i ii i i i i

dE d L L L d L L dq q q q q Q W Wdt dt q q q dt q q dt= = =

∂ ∂ ∂ ∂ ∂= − − = − = = + ∂ ∂ ∂ ∂ ∂

∑ ∑ ∑& & && & && & &

(14)

The above equation simply expresses the energy conservation law for the fluid-structure system: 0M A DE E W W− = + (15)

where 0E is the initial mechanical energy of the deck section at zero reference point, for example, at t1=0. When integrating over one period of vibration, the energy conservation law becomes

M A DE W W∆ = ∆ + ∆ (16) In the above equation, ME∆ , AW∆ and DW∆ denote mechanical energy increment, aerodynamic

work and damping dissipation energy over LCF period. According to Eq. (16), the work done by the aerodynamic forces are partially dissipated by the structural damping, and the remaining work goes into increasing the total mechanical energy of the deck section.

Figure 9 shows the time history of ME∆ , AW∆ and DW∆ at different wind speeds. Let us firstly focus on the divergent flutter case as shown in Figure 9b. ∆ AW remains positive value, which indicates the fluid-structure system always absorbs energy from the air flow during vibration. The value of AW∆ and ∆ DW increase over time. However, the work done by the aerodynamic forces is much larger than the damping dissipation energy, i.e., 0A DW W∆ + ∆ ＞ , which lead to occurrence of divergent vibration and eventual failure.

For the case of LCF, DW∆ increases over time till the vibration reaches steady state. AW∆ also remains positive value during vibration. In the time interval [2, 10], the work done by the aerodynamic forces is larger than the damping dissipation energy; so the deck section experiences divergent vibration during this process. The value of AW∆ first increases then decreases over time, and it reaches a maximum at t≈6s. Although the motion amplitude continues to increase in the time interval [6, 10], the aerodynamic work AW∆ decreases over time. At t≈10s, the total energy absorbed from the air flow are effectively dissipated by the structural damping, i.e., 0M A DE W W∆ = ∆ + ∆ = , which indicates the LCF occurs. Thus, the LCF only occurs if the total mechanical energy of the deck section remains unchanged in a vibration period. From the energy standpoint, we conclude that

0A DW W Limit Cycle Flutter∆ + ∆ ⇔= (17) is a necessary condition for LCF.

2 4 6 8 10 12-0.06

-0.03

0.00

0.03

0.06

0.09

0.12 ∆EM ∆WA ∆WD

Ener

gy (J

)

t (s)0.5 1.0 1.5 2.0 2.5 3.0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0 ∆EM ∆WA ∆WD

Ener

gy (J

)

t (s) (a) U0 = 12 m/s (b) U0 = 13.5 m/s

Figure 9: Time history of ∆ ME , ∆ AW , ∆ DW , ξ0=0.005.

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7 CONCLUSIONS

Based on ANSYS FLUENT, a fluid-structure interaction (FSI) model has been developed to investigate the LCF characteristics of a bridge deck. The numerically simulated critical flutter speed and frequency of a thin plate section exhibit good agreement with theoretical solutions, by which the applicability and accuracy of the numerical approach is validated. The LCF responses of the bridge deck were also successfully simulated. The LCF amplitude gradually increases with wind speed, and eventually the divergent flutter occurs when the wind speed reaches a specific value. The critical wind speed of LCF can be significantly increased by increasing the structural damping. However, the structural damping has negligible effect on the frequency of LCF.

The total mechanical energy varies bounded during LCF, and it remains unchanged in a vibration period. When the wind speed is sufficient high, the work done by the aerodynamic forces is larger than the damping dissipation energy, and the deck section experiences divergent vibration; LCF occurs when the aerodynamic work done by air flow on the deck section is equal to the dissipated energy by structure damping.

REFERENCES

Bendiksen, O.O., Transonic Limit Cycle Flutter/LCO. Proc. Of 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, California, 2004.

Chen, X., Matsumoto, M., Kareem, A., Aerodynamic Coupling Effects on Flutter and Buffeting of Bridge. Journal of Engineering Mechanics, 126(1): 17-26, 2000.

Chen, X., and Kareem, A., Efficacy of tuned mass dampers for bridge flutter control. Journal of Engineering Mechanics, 130(12): 2070-2074, 2004.

Fransos, D., and Bruno, L., Determination of the aeroelastic transfer functions for streamlined bodies by means of a Navier–Stokes solver. Mathematical and Computer Modelling, 43(5), 506-529, 2006.

Hua, X.G., Chen, Z.Q., Full-order and multimode flutter analysis using ANSYS. Finite Elements in Analysis & Design, 44(9): 537–551, 2008.

Körlin, R., Starossek, U., Active mass dampers for flutter control of bridges. Proc. of International Conference on Flow-induced Vibrations, Paris, 2004.

Lee, B.H.K., LeBlanc, P., Flutter analysis of a two-dimensional airfoil with cubic nonli-near restoring force. Aeronautical Note NAE-AN-36, NRC No. 26438, National Research Council Canada, 1986.

Li, K., Ge, Y.J., Guo, Z.W., Zhao, L., Theoretical framework of feedback aerodynamic control of flutter oscillation for long-span suspension bridges by the twin-winglet system. Journal of Wind Engineering & Industrial Aerodynamics, 145: 166-177, 2015.

Liu Z.J., Yang, T.X., Ge, Y.J., Zhang, W., Numerical simulation of box-girders flutter and pneumatic energy analysis. Engineering Mechanics, 32(9): 58-67, 2015. (in Chinese)

Long, F., Wind-resistant performance analysis and experimental investigation on long-span self-anchored suspension bridge. Dalian University of Technology, China, 2010. (in Chinese)

Menter, F. R., Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal, 32(8):1598-1605, 1994.

Xu, F.Y., Chen, A.R., 3-D flutter analysis of Sutong bridge. Engineering Mechanics, 25(8): 139-144, 2008. (in Chinese).

Scanlan, R. H., Tomko, J., Air foil and bridge deck flutter derivatives. Journal of Engineering Mechanics, 97(6), 1717-1737, 1971.

Schneider, G., and Raw, M., Control volume finite-element method for heat transfer and fluid flow using colocated variables—1. computational procedure. Numerical Heat Transfer, Part A Applications, 11(4), 363-390, 1987.

Xu, F. Y., Ying, X. Y., Zhang, Z., Three-degree-of-freedom coupled numerical technique for extracting 18 aerodynamic derivatives of Bridge Decks. Journal of Structural Engineering, 140(11), 2014.

Ying, X.Y., Xu, F.Y., Zhang, Z., Numerical simulation and visualization of flow around rectangular bluff bodies." Proc. of BBAA 7 Int. Colloquium on Bluff Bodies Aerodynamics and Applications, Shanghai, China, 2012.

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Zhang, C.G., Soft flutter and parameters identification of nonlinear self-excited aerody-namic force of bridge girder. Tongji University, China, 2007.

Zhang, C.W., Li, J.L., Li, H., Ou, J.P., Preliminary Numerical Study on TRID System for Flutter Vibration Control of Bridge Structure. Procedia Engineering, 14(2): 2796-2806, 2011.

Zhu, L., Gao, G., Influential factors of soft flutter phenomenon for typical bridge deck sections. Journal of Tongji University (Nautual Science), 43(9): 1219-1294, 2015.. (in Chinese)

Zhang, Q. and Hisada, T., Studies of the strong coupling and weak coupling methods in FSI analysis." International Journal for Numerical Methods in Engineering, 60(12): 2013-2029, 2004.