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.     

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.     

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.     

Anatomy of Turbulence Effects on the Aerodynamics of an Oscillating Prism

Turbulence-induced changes in aerodynamic force characteristics on an oscillating rectangular prism were investigated in this study. It encompassed examination of aeroelastic quantities, i.e., flutter derivatives, and the buffeting components of the integral forces. A forced-vibration system was employed to extract the aerodynamic characteristics of an oscillating prism using a model instrumented with multiple pressure transducers for synchronous scanning of the pressure field. Chordwise distributions of self-excited pressure amplitudes were measured and associated phases were derived to examine the anatomy of turbulence effects on the aerodynamics of the prism rather than simply discern the influence of turbulence on the integrated forces. The resulting changes in flutter derivatives were traced back to a turbulence-induced upstream shifting of the regions of maximum pressure amplitudes. This upstream shifting was consistent with earlier research showing that turbulence increases the radius of curvature of separated shear layers and moves reattachment upstream. In this study, turbulence was found to have a stabilizing effect on the aerodynamics of the prism. The broad band character of the buffeting forces was found to be quite similar to that of stationary prisms with body motion slightly increasing energy content.     

Suppression of Bridge Flutter by Active Deck-Flaps Control System

A theoretical study on an aerodynamic control method for suppression of the wind-induced instabilities of a very long span bridge is presented in this paper. The control system consists of additional control flaps attached to the edges of the bridge deck. Their rotational movement, commanded via feedback control law, is used to modify the aerodynamic forces acting on the deck and provides aerodynamic forces on the flaps used to stabilize the bridge. A time domain formulation of self-excited and buffeting forces is obtained through the rational function approximation of the generalized Theodorsen function. The optimal configuration of the deck-flaps system is found with respect to the performance index based on stability robustness of the system. A control system with the rotational center of the flaps that is located on the edges of the deck was found to be the most effective. It is also shown that this control system can provide sufficient aerodynamic damping and satisfactory stability robustness of the system with a relatively small flap size for the considered range of wind speed.     

Flutter and Buffeting Analysis.?II: Luling and Deer Isle Bridges

The practical applications of the finite-element software developed in Part I of this paper are demonstrated by analyzing the aerodynamic response of two prototype bridges, the Luling Bridge with a streamlined section, and the Deer Isle Bridge with a bluff section. In the Luling Bridge analysis, the self-excited forces are treated as random quantities and the wind turbulence effect is simulated with random parametric excitation analysis. In the Deer Isle Bridge flutter analysis, the self-excited forces are treated as deterministic quantities and the wind turbulence effect is included in the flutter derivatives measured in turbulent wind.     

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.     

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.     

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.     

Transverse Shear Effect on Flutter of Composite Panels

The effect of transverse shear deformation on the supersonic flutter of composite panels has been investigated using the finite element method. First‐order shear‐deformation laminated‐plate theory and quasi‐steady aerodynamic theory are employed for the analysis. The total displacement of the plate is expressed as the sum of the displacement due to bending and the displacement due to shear deformation. Thus, the aerodynamic pressure induced by the plate motion is also the sum of the pressure induced by bending deformation and the pressure induced by shear deformation. Numerical results show that the transverse shear deformation may have a significant effect on the flutter boundary if aerodynamic damping were small or neglected in the determination of flutter boundary.     

Flutter of an Arch Bridge via a Fully Nonlinear Continuum Formulation

A fully nonlinear parametric model for wind-excited arch bridges is proposed to carry out the flutter analysis of Ponte della Musica under construction in Rome. Within the context of an exact kinematic formulation, all of the deformation modes are considered (extensional, shear, torsional, in-plane, and out-of-plane bending modes) both in the deck and supporting arches. The nonlinear equations of motion are obtained via a total Lagrangian formulation while linearly elastic constitutive equations are adopted for all structural members. The parametric nonlinear model is employed to investigate the bridge limit states appearing either as a divergence bifurcation (limit point obtained by path following the response under an increasing multiplier of the vertical accidental loads) or as a Hopf bifurcation of a suitable eigenvalue problem (where the bifurcation parameter is the wind speed). The eigenvalue problem ensues from the governing equations of motion linearized about the in-service prestressed bridge configuration under the dead loads and wind-induced forces. The latter are expressed in terms of the aeroelastic derivatives evaluated through wind-tunnel tests conducted on a sectional model of the bridge. The results of the aeroelastic analysis—flutter speed and critical flutter mode shape—show a high sensitivity of the flutter condition with respect to the level of prestress and the bridge structural damping.     

Time-Domain Model for Predicting Aerodynamic Loads on a Slender Support Structure for Fatigue Design

This paper presents the development of a universal model for predicting cyclic aerodynamic loads originating from buffeting, self-excited, and vortex shedding on a slender support structure in the time domain that can be used to predict its fatigue life. To accomplish this development, long-term monitoring was performed on a high mast light pole (HMLP) and the field data were used to validate the developed mathematical model. Wind-tunnel tests were conducted on the dodecagonal (12-sided) cylindrical cross section of the light pole to obtain the necessary aerodynamic parameters such as static force coefficients, Strouhal number, and indicial functions for buffeting that appear in the postulated model. Furthermore, these aerodynamic parameters were cast into a coupled dynamic model for predicting the response of any HMLP in time domain from vortex shedding and buffeting.     

Control of Wind-Induced Self-Excited Oscillations by Transfer of Internal Energy to Higher Modes of Vibration. I: Analysis in Two Degrees of Freedom

This paper and its companion paper present a new remedy to control wind-induced self-excited oscillation of long and flexible structures with low-internal damping, such as stay cables in cable-stayed bridges. A simple magnetic or mechanical device is used to disturb the cable motion in the lower modes of vibration and thus to transfer a portion of the internal energy of the system from the lower modes to higher modes. Because higher modes generally have high-positive aerodynamic damping when lower modes are excited by wind, the transferred energy is dissipated during the decay of high-frequency vibration. The present paper aims at capturing the fundamentals of energy transfer and dissipation through a detailed theoretical analysis based on a simplified model: A two-degree-of-freedom system with galloping-type self-exciting wind forces. The efficiency to reduce the amplitude of oscillation with a passive device and two types of semiactive devices is demonstrated using energy considerations. Results of numerical simulations are also presented.     

Study of Passive Deck-Flaps Flutter Control System on Full Bridge Model. I: Theory

A passive aerodynamic control method for suppression of the wind-induced instabilities of a very long span bridge is presented in this paper. The control system consists of additional control flaps attached to the edges of the bridge deck. Control flap rotations are governed by prestressed springs and additional cables spanned between the control flaps and an auxiliary transverse beam supported by the main cables of the bridge. The rotational movement of the flaps is used to modify the aerodynamic forces acting on the deck and provides aerodynamic forces on the flaps used to stabilize the bridge. A time-domain formulation of self-excited forces for the whole three-dimensional suspension bridge model is obtained through a rational function approximation of the generalized Theodorsen function and implemented in the FEM formulation. This paper lays the theoretical groundwork for the one that follows.     

Theory and Full-Bridge Modeling of Wind Response of Cable-Supported Bridges

As is well known, long, suspended bridge spans require, in the design stage, careful study of their resistance and response to site winds. This has driven, on the one hand, detailed quantitative observation of bridge models in the wind tunnel and, on the other, a steady development and refinement of parallel theory. Currently, both aspects have arrived at good stages of sophistication, although with continued room for improvement. Successes in the extension of bridge spans to record-breaking lengths are mainly due to progress in wind-resistant design, a primary component in the design of long-span bridges. Recently, multimode flutter and buffeting analysis procedures have been developed. These procedures, which were based centrally on frequency-domain methods, take into account the fully coupled aeroelastic and aerodynamic response of long-span bridges to wind excitation. This paper briefly reviews the current state of the art in long-span bridge wind analysis, emphasizing the analytical infrastructure. The focus then turns to exhibit an example of application of the theory to the stability (flutter) and serviceability (buffeting) analyses of a new long-span bridge in North America. This example not only demonstrates the application of the theory to a real structure but also serves to highlight some insights into the versatility that is gained by this analytically based approach. The results demonstrate that the analytical method with appropriate inputs and a complementary full-bridge model agree even for relatively unusual incoming turbulence in the flow caused by the presence of structures upstream of the bridge. This paper seeks to exhibit recent developments in the field to the interested structural∕bridge engineer, outline alternative procedures available for assessment of wind effects on cable-supported bridges, and provide an overview of the basic steps in the process of a typical aerodynamic analysis and design.     

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.     

Nonlinear Aeroelastic Simulation of a Full-Span Aircraft with Oscillating Control Surfaces

In this paper, the transonic and low-supersonic aeroelastic behavior of the generic fighter model was investigated in the time domain. The simulation of flutter flight test using forced harmonic motion of control surfaces including inertial coupling effects was conducted at the various conditions. The detailed dynamic aeroelastic responses are computed using a coupled time-marching method based on the effective computational structural dynamic and computational fluid dynamics techniques. The nonlinear aerodynamic effects due to an existing shock wave on the lifting surfaces were considered using a transonic small disturbance equation. A modal model obtained by a free vibration analysis was used for the structural model. The relations between the computed flutter boundary and the simulation results of the responses using the harmonic motions of control surfaces at various conditions were investigated.