Time Domain Calculation of Vortex Induced Vibration of Long-Span Bridges by Using a Reduced-order Modeling Technique

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1 2017 2nd International Conference on Industrial Aerodynamics (ICIA 2017) ISBN: Time Domain Calculation of Vortex Induced Vibration of Long-San Bridges by Using a Reduced-order Modeling Technique Kun Xu, Lin Zhao, Yaojun Ge and Xiuli Du ABSTRACT A time domain strategy for calculating the vortex-induced vibration (VIV) of bridges is roosed in this study. The strong couling of fluid-structure interactions during VIV is modeled by using a reduced order modeling (ROM) technique. Then the rototye VIV resonse of the bridge is calculated through combing the reduced order model with the finite element method. A real bridge is involved as a case study and is validated through data from field measurements. The roosed strategy is indeendent of the structural mass-daming arameter and the influence of actual flow conditions, for examle slowly time-varying wind velocity during VIV, can also be taken into account in the calculation, which is far beyond the caacity of traditional semi-emirical models. INTRODUCTION Vortex-induced vibration widely exists in many engineering fields, such as longsan bridges, chimneys, and marine structures. Due to its significant influence on the usability and fatigue life of the structures, carefully attentions should be aid to its rediction in the concet design stage. Kun Xu, Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education, Beijing University of Technology, Beijing, China. Lin Zhao, State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China. Yaojun Ge, State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China. Xiuli Du, Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education, Beijing University of Technology, Beijing, China. 825

2 Due to the comlexities existing in the fluid-structure interactions during VIV, there exists no any analytic formula that theoretically describes this henomenon. And since the involving of a full bridge in the CFD aroach with three-dimensional grids can hardly be imlemented in a foreseeable future, the analysis of VIV of bridges at resent is conducted based on emirical models with arameters determined through wind tunnel exeriments. The first attemt to model the VIV of bridge decks was made by Scanlan [1], who roosed a van der ol tye equation that governs the motion of the deck. Since then, several attemts have been made in the same direction, where the hysical mechanism of energy transfer from the flow to the deck during VIV is considered to be an instability of the negative daming of the total VIV system [2-4]. However, these kinds of models are deendent on a structural arameter, the mass-daming arameter. If the mass-daming arameter of a structure changes, the aerodynamic arameters in these model also vary. Thus, the semi-emirical models fail in the rediction for a certain deck shae for values of mass-daming arameters different from that for which the model arameters were estimated. As to the calculation of rototye VIV resonses of a bridge, there exist two main categories. The first is to derive the element aeroelastic matrices through Hamilton s rincile and assemble the element aeroelastic matrices with element inertial, elastic and daming matrices to form the equation of motion for the whole VIV system [5]. The other strategy is based on the modal exansion the structure, where the VIV system is treated as a generalized single-degree-of-freedom (SDOF) nonlinear system that has the same modal roerties with a certain mode of the structure [6]. Since the generation of element aeroelastic matrices or the SDOF nonlinear equations is based on the traditional semi-emirical models, the calculated rototye VIV resonse of the bridge is limited by the caacity of the semi-emirical models. Recently, some concetually novel and comutationally efficient techniques have been roosed for comuting unsteady flows and fluid-induced effects. Among the various ROMs, the Volterra theory-based model rovides the ossibility to catch the inherent nature of nonlinearities and memory effects existing in fluid-structure interactions. Wu and Kareem [7] introduced it into bridge wind engineering to model aerodynamics and aeroelasticity. Xu et al [8] studied the caacity of Volterra series in modeling the strong couling character of fluid-structure interactions during VIV. In this aer, the strategy for calculating the rototye VIV resonse of the bridge by combing ROM with the finite element method is roosed. A real bridge is involved as a case study and the calculated results are validated through data from field measurements. 826

3 VOLTERRA SERIES MODELING OF VIV The VIV of a bridge deck often occurs in a single degree of freedom. Taking the vertical degree of freedom as an illustration, the VIV system can tyically be treated as time-invariant, causal and with fading memory. For such a system, the lift force can be uniformly reresented as: where, L t, M t t t h (,, ) y( t ) y( t ) d d t t C t h y t d h y t y t d d h1 is the first-order kernel, which describes the linear behavior of the system; h( 1,, ) is the th -order kernel term, which describes the th -order nonlinearity existing in the system; and M is the degree of nonlinearity to be considered in the VIV system. In real ractice, the time series of both yt () and t generated either from exeriments or CFD calculations are discrete time series. Thus, Eq. (1) is always written in discrete-time form, which can be exressed as: where L M 1 CL (1) C k H k (2) M M (3) H k h m,, m y k m 1 j m1 0 m 0 j 1 The kernels in the above equations are in fact linear and nonlinear transfer functions between structural motion and aeroelastic forces, which describes the couling effect of fluid-structural interactions during VIV and its duration time. Thus, an accurate identification of these kernels is of vital imortance in the modeling fluid-structural interaction. Xu et al [8] studied the influences of the degree of nonlinearity M and the duration time M on the identification efficiency and accuracy of the kernel terms. And the full and sarse form kernels are also comared in this study. The results indicate that the modeling erformance of a higher-order Sarse Volterra model is better than that of a lower-order full Volterra model. A recommendation on the choice of duration time M is also made in this study, where the first- and secondorder terms 1 and 2 are set to 20 and 10 non-dimensional times, resectively, And for the other higher-order terms, the non-dimensional duration time ( 3) are set to 5 to sufficiently take into account the higher order nonlinear effects existing in the VIV of bridge decks. 827

4 CALCULATION OF PROTOTYPE RESPONSE OF A BRIDGE The rototye bridge is a flexible structure with 3D modal shaes, while the sectional model in wind tunnel is a rigid structure. Thus, the VIV resonse of a sectional model varies with that of a real bridge. To calculate the rototye resonse of a bridge, the VIV models should be combined with the finite element method. In this study, the Volterra model is utilized to model the VIV forces on each segment of the bridge deck along the san-wise, and these VIV forces are then alied on the node of each structural finite element. The strategy of calculation consists of the following stes: Firstly, the initial condition is given, which is generally selected to be a small value of structural amlitude or velocity, which is satially distributed along the san-wise direction following the modal shae of a selected mode. Then, the VIV forces at each san-wise segment of the bridge are generated through the ROM aroach with the given initial condition, and are alied on the structural finite element model to calculate the first ste of structural motions. The calculated structural motions are then fed back to the ROM aroach to generate the next ste of VIV forces on each segment of the structure along the san-wise direction. By iterating the above stes, the bridge s resonse can be generated ste by ste, and the VIV erformance of a bridge can be obtained after calculating the VIV resonse at different wind velocities for each mode. The calculation of structural motion at each iteration ste can be fulfilled in the hysical coordinate system, where the san-wise VIV forces are directly alied on the structural finite element model, or conducted in the modal coordinate system, since the bridge s VIV often occurs in a single mode. When conducting the calculation in modal coordinates, the modal generalized force should be generated at each time ste. The technical roadma for calculating the bridge s VIV is illustrated by Figure 1. Start VIV forces at each san-wise osition Initial condition Reduced order modeling Structural motion of each segment along the san-wise Physical coordinate to modal coordinate Equations of motion Structural motion Modal coordinate to hysical coordinate Figure 1. Technical roadma for bridge VIV calculation. 828

5 Due to the three-dimensional effects of flow structures, the VIV forces are not fully correlated along the san-wise. To obtain an accurate rototye VIV resonse, the influence of satial coherence of VIV forces should be taken into account in the calculation. According to Ehsan and Scanlan [6], the influence of satial coherence of VIV force can be taken into account through two different methods. The first is an exonential tye of satial coherence function that is a function of structural amlitude and satial distance between two deck segments. The rototye VIV amlitude of a bridge can be obtained through an iteration algorithm. The other is the correction function method, where the generalized VIV force for a certain mode of the bridge is modified by a correction function. The correction function is used to take the non-fully correlation of VIV forces along the san-wise direction into account. According to the roadma of Figure 1, the solution of the equation of motion can be conducted in both the hysical and/or modal coordinates. If the calculation of the equation of motion is conducted in the modal coordinate at each time ste, the correction function method can be adoted to take the non-fully correlation of VIV forces into account. Thus, the correlation function method is adoted in the calculation of rototye VIV resonse of the bridge. The modified generalized VIV force for the ith mode of the bridge can be exressed as: ˆ L ( ) ( ) ( ) dx (4) Fi F 0 i x g x x L where Fi ( x ) is the VIV force at san-wise location of x, which is modeled by the Volterra series; () x is modal shae function of the deck; L is the total length of the deck; gx ( ) is the correction function with a maximum value of unit. The correction function can be obtained through wind tunnel exeriment on sectional or aeroelastic models, or roughly adot the normalized modal shae of the ith mode of the bridge, as done with Ehsan and Scanlan [6]. CASE STUDY To illustrate and validate the above established rocedure, the Xihoumen Bridge is adoted here as a case study. The Xihoumen Bridge is a susension bridge with a main san of 1650m. The bridge deck consists of two boxes with a central slot of 6m. This bridge exerienced single mode VIV in several vertical bending modes at moderate wind seeds in the range of 5m/s~13m/s after erection but before being oened to the ublic[9]. Since the bridge exerienced vertical bending mode VIV, the vertical bending mode VIV are calculated in this case study. The modal roerties of the vertical bending modes among the first 100 modes of the Xihoumen Bridge are illustrated by Table I. 829

6 Modal number TABLE I. MODAL PROPERTIES OF THE VERTICAL BENDING MODES AMONG THE FIRST 100 MODES. Equivalent modal Frequency Modal Frequency mass of the deck (Hz) number (Hz) (Kg/m) Equivalent modal mass of the deck (Kg/m) E E E E E E E E E E E E E e E E E E E E E E E E E E E E E E E+04 VIV resonse at fixed wind velocities The strong couling characteristic of fluid structure interactions during VIV are modeled by using the Volterra series. The kernels in the Volterra series are identified through data obtained from a series of forced vibration with different amlitudes and frequencies. The details on the kernel identification of the Xihoumen Bridge and its validation can be found in Xu et al[8]. After a comrehensive study on the identification efficiency and accuracy, a 7-order sarse form Volterra model is adoted to simulate the fluid-structural interactions of this deck shae during VIV. After obtaining the kernels in the Volterra series, the rototye VIV resonse of the bridge can be obtained through the method roosed in the above section. The calculated steady-state maximum VIV resonses of the first 19 th vertical bending modes of the Xihoumen Bridge are illustrated by Figure 2. The influence of the satial coherence of VIV forces are taken into account through the correction function method. Due to the lack of essential exerimental data on the satial coherence of VIV forces for this deck shae, the normalized shae function of each mode is taken as the correction function of this bridge in the calculation of the rototye VIV resonse of the bridge. As is indicated by Figure 2, the calculated VIV amlitudes are qualitatively comarable with those of field measurement. However, the calculated velocity range of lock-in is a little bit smaller than that of field measurement for each mode. This may be due to the inconsistence between the CFD aroach adoted in this study for kernel identification and the actual condition. Since the deck shae simulated in the 830

7 Reduced Amlitude CFD aroach does not have railings while the actual deck does. Moreover, the calculation domain in the CFD aroach is two-dimensional, which cannot consider the three-dimensionality of the flow fields and the influences of discretely distributed cross beams used for connecting the twin boxes Calculated Results: 6th mode 7th mode 9th mode 10th mode Field Measurements: 6th mode 7th mode 9th mode 10th mode Wind Velocity (m/s) Figure 2. Calculated steady-state VIV amlitudes and those of field measurements. VIV resonse at slowly-varying velocities Field measurements indicate that the actual oncoming flow velocities during VIV are not constant but slowly varying with time. The traditional semi-emirical models deend on the reduced wind velocity, thus cannot take the influence of actual flow roerties into account. For some suer long-san bridges, active aerodynamic control measures may have to be adoted to mitigate the VIV resonse of the bridge. The control devices need to change their attitudes in tune with the vibration status of the structure. In this case, the real-time VIV resonse of the bridge should be calculated and the influence of actual flow roerties has to be considered. The VIV resonse of the bridge at slowly-varying wind velocities are calculated by using the time domain aroach in this study. According to the results of field measurements, the Xihoumen Bridge exerienced the 7 th vertical bending mode vibration in the wind velocity range of 6.4m/s~7m/s. Due to the lack of the time history of wind velocities on site, the time history of wind velocity in the calculation is modeled by an assumed form, where the wind velocity is slowly growing from the most unfavorable wind velocity to exceed the lock-in region for the 7 th vertical bending mode of this bridge. The slowly-varying wind velocity is defined as follows: cos(3 2 t / 2 t ) 2 U end (5) where, 6.28m/s is around most unfavorable wind velocity for the 7 th vertical bending mode of this bridge. The end value of this velocity is U=7.43m/s. The time 831

8 Y=y/D U(s) interval of the wind velocity function should be equal to that for the solution of the equation of motion. Figure 3 shows the calculated results and those of field measurements. As is illustrated by this figure, the VIV amlitude of the bridge slowly decreases with the increasing of wind velocity, which shows a similar vibration characteristic with that of field measurements U=6.43~7.01m/s [9] s=ut/d Figure 3. Calculated dislacement of the bridge and that of field measurement. Another examle of slowly-varying wind velocity is illustrated by the flowing equation: s cos(3 2 ( s 5000) /14000) 5000 s Us () 2 (6) cos( ( s 12000) / 6000) s s The wind velocity slowly grows from the outside of lock-in region to the most unfavorable wind velocity and then raidly decreases to the outside of the lock-in region. The beginning wind velocity is U 5.75 m / s and the maximum value of wind velocity is Uh 6.5 m / s. The calculated results and those of field measurements are illustrated by Figure 4. The VIV amlitude of the bridge slowly grows with the increase of wind velocity and then raidly decreases with the decrease of wind velocity. The vibration characteristic of the bridge comare well with that of field measurements, which validates the feasibility of this time domain calculation strategy. i 832

9 U(s) Y=y/D U=8.31~8.89m/s [9] s=ut/d Figure 4. Calculated dislacement of the bridge and that of field measurement. CONCLUSIONS A time domain strategy for calculating the vortex-induced vibration (VIV) of bridges by using a reduced order modeling technique is roosed in this research. This strategy is indeendent of the structural mass-daming arameter and the reduced wind velocity. Thus, the influence of actual flow conditions, for examle slowly time-varying wind velocity during VIV, can be taken into account in the calculation. A real bridge is adoted as a case study and the calculated results comare well with those of field measurements. The erformance of this strategy is far beyond the caacity of traditional semi-emirical models. This method can also be alied to other VIV sensitive structures, such as chimney or marine structures. REFERENCES Scanlan, R.H., On the state-of-the-art methods for calculations of flutter, vortex-induced and buffeting resonse of bridge structures. 1981: FHWA-RD Final Reort. Larsen, A., A generalized model for assessment of vortex-induced vibrations of flexible structures. Journal of Wind Engineering and Industrial Aerodynamics, (2): Zhu, L.D., X.L. Meng, and Z.S. Guo, Nonlinear mathematical model of vortex-induced vertical force on a flat closed-box bridge deck. Journal of Wind Engineering & Industrial Aerodynamics, (11): Xu, K., Y. Ge, and D. Zhang, Wake oscillator model for assessment of vortex-induced vibration of flexible structures under wind action. Journal of Wind Engineering & Industrial Aerodynamics, : Barhoush, H., A.H. Namini, and R.A. Sko, Vortex shedding analysis by finite elements. Journal of Sound & Vibration, (1): Ehsan, F. and R.H. Scanlan, Vortex-induced vibrations of flexible bridges. Journal of Engineering Mechanics, (6): Wu, T. and A. Kareem, A nonlinear analysis framework for bluff-body aerodynamics: A Volterra reresentation of the solution of Navier-Stokes equations. Journal of Fluids & Structures, :

10 8. Xu, K., L. Zhao, and Y. Ge, Reduced-order modeling and calculation of vortex-induced vibration for large-san bridges. Journal of Wind Engineering & Industrial Aerodynamics, : Li, H., et al., Investigation of vortex-induced vibration of a susension bridge with two searated steel box girders based on field measurements. Engineering Structures, (6):