Aerodynamic Coupling Effects on Flutter and Buffeting of Bridges

The effects of aerodynamic coupling among modes of vibration on the flutter and buffeting response of long-span bridges are investigated. By introducing the unsteady, self-excited aerodynamic forces in terms of rational function approximations, the equations of motion in generalized modal coordinates are transformed into a frequency-independent state-space format. The frequencies, damping ratios, and complex mode shapes at a prescribed wind velocity, and the critical flutter conditions, are identified by solving a complex eigenvalue problem. A significant feature of this approach is that an iterative solution for determining the flutter conditions is not necessary, because the equations of motion are independent of frequency. The energy increase in each flutter motion cycle is examined using the work done by the generalized aerodynamic forces or by the self-excited forces along the bridge axis. Accordingly, their contribution to the aerodynamic damping can be clearly identified. The multimode flutter generation mechanism and the roles of flutter derivatives are investigated. Finally, the coupling effects on the buffeting response due to self-excited forces are also discussed.     

Time Domain Flutter and Buffeting Response Analysis of Bridges

A time domain approach for predicting the coupled flutter and buffeting response of long span bridges is presented. The frequency dependent unsteady aerodynamic forces are represented by the convolution integrals involving the aerodynamic impulse function and structural motions or wind fluctuations. The aerodynamic impulse functions are derived from experimentally measured flutter derivatives, aerodynamic admittance functions, and spanwise coherence of aerodynamic forces using rational function approximations. A significant feature of the approach presented here is that the frequency dependent characteristics of unsteady aerodynamic forces and the nonlinearities of both aerodynamic and structural origins can be modeled in the response analysis. The flutter and buffeting response of a long span suspension bridge is analyzed using the proposed time domain approach. The results show good agreement with those from the frequency domain analysis. The example used to demonstrate the proposed scheme focuses on the treatment of frequency dependent self-excited and buffeting force effects. Application to nonlinear effects will be addressed in a future publication.     

Dynamic Performance Simulation of Long-Span Bridge under Combined Loads of Stochastic Traffic and Wind

Slender long-span bridges exhibit unique features which are not present in short and medium-span bridges such as higher traffic volume, simultaneous presence of multiple vehicles, and sensitivity to wind load. For typical buffeting studies of long-span bridges under wind turbulence, no traffic load was typically considered simultaneously with wind. Recent bridge/vehicle/wind interaction studies highlighted the importance of predicting the bridge dynamic behavior by considering the bridge, the actual traffic load, and wind as a whole coupled system. Existent studies of bridge/vehicle/wind interaction analysis, however, considered only one or several vehicles distributed in an assumed (usually uniform) pattern on the bridge. For long-span bridges which have a high probability of the presence of multiple vehicles including several heavy trucks at a time, such an assumption differs significantly from reality. A new “semideterministic” bridge dynamic analytical model is proposed which considers dynamic interactions between the bridge, wind, and stochastic “real” traffic by integrating the equivalent dynamic wheel load (EDWL) approach and the cellular automaton (CA) traffic flow simulation. As a result of adopting the new analytical model, the long-span bridge dynamic behavior can be statistically predicted with a more realistic and adaptive consideration of combined loads of traffic and wind. A prototype slender cable-stayed bridge is numerically studied with the proposed model. In addition to slender long-span bridges which are sensitive to wind, the proposed model also offers a general approach for other conventional long-span bridges as well as roadway pavements to achieve a more realistic understanding of the structural performance under probabilistic traffic and dynamic interactions.     

Geometrically Nonlinear Buffeting Response of a Cable-Stayed Bridge

A geometrically nonlinear buffeting analysis of a cable-stayed bridge in the time domain is described. The bridge structure is modeled with three-dimensional thin-walled beam elements and three-dimensional elastic catenary cable elements. Spatially correlated wind velocity fluctuations are modeled and simulated using an algorithm for generating sample functions of a stationary, multivariate stochastic process according to its prescribed cross-spectral density matrix. Aerodynamic damping and aerodynamic stiffness are formulated based on experimentally determined flutter derivatives. The focus of this paper is on the effect of fluctuating components of the spatially correlated wind velocity on the geometrically nonlinear buffeting response for an 870 m cable-stayed bridge.     

Aeroelastic Analysis of Bridges under Multicorrelated Winds: Integrated State-Space Approach

In this paper, an integrated state-space model of a system with a vector-valued white noise input is presented to describe the dynamic response of bridges under the action of multicorrelated winds. Such a unified model has not been developed before due to a number of innate modeling difficulties. The integrated state-space model is realized based on the state-space models of multicorrelated wind fluctuations, unsteady buffeting and self-excited aerodynamic forces, and the bridge dynamics. Both the equations of motion at the full order in the physical coordinates and at the reduced-order in the generalized modal coordinates are presented. This state-space model allows direct evaluation of the covariance matrix of the response using the Lyapunov equation, which presents higher computational efficiency than the conventional spectral analysis approach. This state-space model also adds time domain simulation of multicorrelated wind fluctuations, the associated unsteady frequency dependent aerodynamic forces, and the attendant motions of the structure. The structural and aerodynamic coupling effects among structural modes can be easily included in the analysis. The model also facilitates consideration of various nonlinearities of both structural and aerodynamic origins in the response analysis. An application of this approach to a long-span cable-stayed bridge illustrates the effectiveness of this scheme for a linear problem. An extension of the proposed analysis framework to include structural and aerodynamic nonlinearities is immediate once the nonlinear structural and aerodynamic characteristics of the bridge are established.     

Flutter and Buffeting Analysis.?I: Finite-Element and RPE Solution

A finite-element formulation is developed for analyzing flutter instability and buffeting response of long-span bridges and their interaction. The flutter derivatives, instead of the indicial functions used by previous researchers, are applied in the random parametric excitation (RPE) analysis. This application makes finite-element formulation possible and results in much less computational effort in RPE analysis than those of previous analyses. With the finite-element program developed in the present study, as many modes as desired can be easily included in the flutter and buffeting analyses. Users have the choice of RPE or eigenvalue method for flutter analysis and RPE or spectral method for buffeting analysis.     

Aeroelastic Analysis of Bridges: Effects of Turbulence and Aerodynamic Nonlinearities

Current linear aeroelastic analysis approaches are not suited for capturing the emerging concerns in bridge aerodynamics introduced by aerodynamic nonlinearities and turbulence effects. These issues may become critical for bridges with increasing spans and/or with aerodynamic characteristics sensitive to the effective angle of incidence. This paper presents a nonlinear aerodynamic force model and associated time domain analysis framework for predicting the aeroelastic response of bridges under turbulent winds. The nonlinear force model separates the aerodynamic force into low- and high-frequency components according to the effective angle of incidence. The low-frequency force component is modeled utilizing quasi-steady theory. The high-frequency force component is based on the frequency dependent unsteady aerodynamic characteristics, which are similar to the traditional force model but vary in space and time following the low-frequency effective angle of incidence. The proposed framework provides an effective analysis tool to study the influence of structural and aerodynamic nonlinearities and turbulence on the bridge aeroelastic response. The effectiveness of this approach is demonstrated by utilizing an example of a long span suspension bridge with aerodynamic characteristics sensitive to the angle of incidence. The influence of mean wind angle of incidence on the aeroelastic modal properties and the associated aeroelastic response and the sensitivity of bridge response to nonlinear aerodynamics and low-frequency turbulence are examined.     

Comparison of Ambient Vibration Response of the Runyang Suspension Bridge under Skew Winds with Time-Domain Numerical Predictions

Existing buffeting theories and methods for dealing with skew winds are summarized. A modified version of the mean wind decomposition method is described that directly uses wind monitoring data from structural health monitoring systems. A method for time-domain buffeting analysis is presented and implemented in ANSYS, in which self-excited forces are modeled as elemental aeroelastic stiffness and damping matrices by element Matrix27. The entire numerical procedure is implemented in MATLAB and ANSYS, and the buffeting responses can be conveniently computed with very concise input data. This method is applied to the buffeting analysis of the Runyang Suspension Bridge during Typhoon Matsa. The measured data of wind characteristics are used to predict the buffeting responses of the bridge with a finite-element model. The predicted buffeting responses are compared with those from field measurements. A reasonably good agreement between the calculation and the measurement validates the effectiveness of the method. Finally, the characteristics and mechanism of buffeting responses of long-span bridges are discussed to provide references for further studies.     

Revisiting Multimode Coupled Bridge Flutter: Some New Insights

Better understanding of the bimodal coupled bridge flutter involving fundamental vertical bending and torsional modes offers valuable insight into multimode coupled flutter, which has primarily been the major concern in the design of long span bridges. This paper presents a new framework that provides closed-form expressions for estimating modal characteristics of bimodal coupled bridge systems and for estimating the onset of flutter. Though not intended as a replacement for complex eigenvalue analysis, it provides important physical insight into the role of self-excited forces in modifying bridge dynamics and the evolution of intermodal coupling with increasing wind velocity. The accuracy and effectiveness of this framework are demonstrated through flutter analysis of a cable-stayed bridge. Based on this analysis scheme, the role of bridge structural and aerodynamic characteristics on flutter, which helps to better tailor the structural systems and deck sections for superior flutter performance, is emphasized. Accordingly, guidance on the selection of critical structural modes and the role of different force components in multimode coupled flutter are delineated. The potential significance of the consideration of intermodal coupling in predicting torsional flutter is highlighted. Finally, clear insight concerning the role of drag force to bridge flutter is presented.     

Equivalent Wheel Load Approach for Slender Cable-Stayed Bridge Fatigue Assessment under Traffic and Wind: Feasibility Study

In the current AASHTO LRFD specifications, the fatigue design considers only one design truck per bridge with 15% dynamic allowance. While this empirical approach may be practical for regular short and medium span bridges, it may not be rational for long-span bridges (e.g., span length >152.4?m or 500?ft) that may carry many heavy trucks simultaneously. Some existent studies suggested that fatigue may not control the design for many small and medium bridges. However, little research on the fatigue performance of long-span bridges subjected to both wind and traffic has been reported and if fatigue could become a dominant issue for such a long-span bridge design is still not clear. Regardless if the current fatigue design specifications are sufficient or not, a real understanding of the traffic effects on bridge performance including fatigue is desirable since the one truck per bridge for fatigue design does not represent the actual traffic condition. As the first step toward the study of fatigue performance of long-span cable-stayed bridges under both busy traffic and wind, the equivalent dynamic wheel load approach is proposed in the current study to simplify the analysis procedure. Based on full interaction analyses of a single-vehicle–bridge–wind system, the dynamic wheel load of the vehicle acting on the bridge can be obtained for a given vehicle type, wind, and driving condition. As a result, the dimension of the coupled equations is independent of the number of vehicles, through which the analyses can be significantly simplified. Such simplification is the key step toward the future fatigue analysis of long-span bridges under a combined action of wind and actual traffic conditions.     

Theories of Aerodynamic Forces on Decks of Long Span Bridges

Long span bridges are one of the most challenging kinds of structures in civil engineering. Wind loading and wind effects are highly important aspects when designing this typology. The interaction between wind and structure, studied by using aeroelasticity theory, allows us to understand several classes of structural instabilities that may appear. Also, wind tunnel data, obtained by conducting careful testing of reduced models of bridges, produce useful information about prototypes' characteristics. A fundamental aspect of bridge design under aeroelastic constraints is identification of aerodynamic forces; several models for this purpose are presented in this paper. First, a model based on a two-degrees-of-freedom plane plate moving in an incompressible fluid is reviewed; this approach, although useful in airfoil engineering, is not valid any longer in civil engineering, as bridge decks are bluff bodies. Second, a linearized theory, also based on a two-degrees-of-freedom model is analyzed; in this case, obtaining aerodynamic forces requires identification of a set of coefficients, called flutter derivatives, that can be found by carrying out testing of reduced models of a segment of bridge deck. Finally, an extension of that approach, leading to a linearized theory of a three-degrees-of-freedom model is presented.     

Dynamic Response of Suspension Bridge to High Wind and Running Train

A framework is presented for predicting the dynamic response of long suspension bridges to high winds and running trains. A three-dimensional finite-element model is used to represent a suspension bridge. Wind forces acting on the bridge, including both buffeting and self-excited forces, are generated in the time domain using a fast spectral representation method and measured aerodynamic coefficients and flutter derivatives. Each 4-axle vehicle in a train is modeled by a 27-degrees-of-freedom dynamic system. The dynamic interaction between the bridge and train is realized through the contact forces between the wheels and track. By applying a mode superposition technique to the bridge only and taking the measured track irregularities as known quantities, the number of degrees of freedom of the bridge-train system is significantly reduced and the coupled equations of motion are efficiently solved. The proposed formulation is then applied to a real wind-excited long suspension bridge carrying a railway inside the bridge deck of a closed cross section. The results show that the formulation presented in this paper can predict the dynamic response of the coupled bridge-train systems under fluctuating winds. The extent of interaction between the bridge and train depends on wind speed and train speed.     

Estimates of Skew Wind Speeds for Bridge Flutter

When estimating the stability of a long-span bridge under wind, the basic study is most often made by considering the wind as it approaches the bridge at right angles to its long axis. However, maximum wind at a given site seldom approaches exactly normal to this axis, but will generally be skew instead. A common assumption is that a given skew wind velocity will be critical for flutter if its cosine component normal to the bridge deck equals the critical bridge normal-wind velocity. Although this does not provide a completely inaccurate estimate, the latter can be sharpened somewhat by considering an available physical approximation to the aeroelastic wind-structure interaction under the skew wind. This approximation is derivable from the experimental analysis of the wind-normal flutter condition and its associated key flutter derivatives, particularly the one linked to deck torsional instability.     

Curve Veering of Eigenvalue Loci of Bridges with Aeroelastic Effects

The eigenvalues of bridges with aeroelastic effects are commonly portrayed in terms of a family of frequency and damping loci as a function of mean wind velocity. Depending on the structural dynamic and aerodynamic characteristics of the bridge, when two frequencies approach one another over a range of wind velocities, their loci tend to repel, thus avoiding an intersection, whereas the mode shapes associated with these two frequencies are exchanged in a rapid but continuous way as if the curves had intersected. This behavior is referred to as the curve veering phenomenon. In this paper, the curve veering of cable-stayed and suspension bridge frequency loci is studied. A perturbation series solution is utilized to estimate the variations of the complex eigenvalues due to small changes in the system parameters and establish the condition under which frequency loci veer, quantified in terms of the difference between adjacent eigenvalues and the level of mode interaction. Prior to the discussion of bridge frequency loci, the curve veering of a two-degree-of-freedom system comprised of a primary structure and tuned mass damper is discussed, which not only provides new insight into the dynamics of this system, but also helps in understanding the veering of bridge frequency loci. To study this more complicated dynamic system, a closed-form solution of a two-degree-of-freedom coupled flutter is obtained, and the underlying physics associated with the heaving branch flutter is discussed in light of the veering of frequency loci. It is demonstrated that the concept of curve veering in bridge frequency loci provides a correct explanation of multimode coupled flutter analysis results for long span bridges and helps to improve understanding of the underlying physics of their aeroelastic behavior.     

Suspension Bridge Flutter for Girders with Separate Control Flaps

Active vibration control of long span suspension bridge flutter using separate control flaps (SFSC) has shown to increase effectively the critical wind speed of the bridges. In this paper, an SFSC calculation based on modal equations of the vertical and torsional motions of the bridge girder including the flaps is presented. The length of the flaps attached to the girder, the flap configuration, and the flap rotational angles are parameters used to increase the critical wind speed of the bridge. To illustrate the theory a numerical example is shown for a suspension bridge of 1,000 + 2,500 + 1,000 m span based on the Great Belt Bridge streamlined girder.     

Effect of Asynchronous Earthquake Motion on Complex Bridges. II: Results and Implications on Assessment

Nonuniform seismic excitation has been shown through previous analytical studies to adversely affect the response of long-span bridge structures. To further understand this phenomenon, this study investigates the response of complex straight and curved long-span bridges under the effect of parametrically varying asynchronous motion. The generation process and modeling procedures are presented in a companion paper. A wide-ranging parametric study is performed aimed at isolating the effect of both bridge curvature and the two main sources of asynchronous strong motion: geometric incoherence and the wave-passage effect. Results from this study indicate that response for the 344?m study structure is amplified significantly by nonsynchronous excitation, with displacement amplification factors between 1.6 and 3.4 for all levels of incoherence. This amplification was not constant or easily predicable, demonstrating the importance of inelastic dynamic analysis using asynchronous motion for assessment and design of this class of structure. Additionally, deck stiffness is shown to significantly affect response amplification, through response comparison between the curved and an equivalent straight bridge. Study results are used to suggest an appropriate domain for consideration of asynchronous excitation, as well as an efficient methodology for analysis.     

Tuned Mass Dampers for Dual-Mode Buffeting Control of Bridges

The objective of this paper is to present the idea of control of the dual-mode buffeting response in suspension bridges under service conditions using two tuned mass dampers (TMDs). The bridge is assumed to vibrate in two vertical modes, which are to be controlled using two TMDs. The TMDs’ properties are determined by minimizing the maximum buffeting response of the bridge. By neglecting modal coupling terms, some approximate formulas are derived. A numerical example is used to understand the requirements for the TMDs. The results show that, as compared to using only one TMD, the dual TMDs offer a more pronounced response reduction. The results also reveal some disadvantages of using the TMD system for buffeting vibration control. The major one is that the efficiency of the TMD system declines and the strokes increase significantly as the wind speed increases.     

Application of Suspension Bridges Stiffened by Prestressed Concrete Slabs in China

Four suspension bridges stiffened by prestressed concrete slabs were designed and constructed on highways in southwestern mountainous areas of China. These bridges are the first applications of its kind in China. This paper discusses the site condition, adaptability, and design and construction features of these bridges. These bridges have single suspension spans between 278 and 388?m and deck width between 14.4 and 15.0?m. The longitudinal distance between hangers is only 5?m, which is relatively small for this bridge type, and there are only two lanes. The dual direction prestressed concrete slabs are 0.6?m deep, and its wind blocking area is relatively small. Dynamic analysis and wind tunnel tests verify that the wind resistance requirements are easily satisfied.