Dynamic Wind Effects on Buildings with 3D Coupled Modes: Application of High Frequency Force Balance Measurements

Contemporary high-rise buildings with complex geometric profiles and three-dimensional (3D) coupled mode shapes often complicate the use of high frequency force balance (HFFB) technique customarily used in wind tunnel testing for uncoupled buildings. In this study, a comprehensive framework for the coupled building response analysis and the modeling of the associated equivalent static wind loads using the HFFB measurement is presented. This includes modeling of building structural systems whose mass centers at different floors may not be located on a single vertical axis. The building response is separated into the mean, background, and resonant components, which are quantified by modal analysis involving three fundamental modes in two translational and torsional directions. The equivalent static wind load is described in terms of the modal inertial loads. The proposed framework takes into account the cross correlation of wind loads acting in different primary directions and the intermodal coupling of modal responses with closely spaced frequencies. Wind load combination is revisited in the context of modeling of the equivalent static wind loads. A representative tall building with 3D coupled modes and closely spaced frequencies is utilized to demonstrate the proposed framework and to highlight the significance of cross correlation of wind loads and the intermodal coupling of modal responses on the accurate prediction of coupled building response. Additionally, delineation of the proper role of the correlation between integrated loads, modal response, and respective building response components in the evaluation of wind effects on coupled buildings is underscored.     

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.     

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.     

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.     

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.     

Vibration Studies of Tsing Ma Suspension Bridge

A three-dimensional dynamic finite element model is established for the Tsing Ma long suspension Bridge in Hong Kong. The two bridge towers made up of reinforced concrete are modeled by three-dimensional Timoshenko beam elements with rigid arms at the connections between columns and beams. The cables and suspenders are modeled by cable elements accounting for geometric nonlinearity due to cable tension. The hybrid steel deck is represented by a single beam with equivalent cross-sectional properties determined by detailed finite element analyses of sectional models. The modal analysis is then performed to determine natural frequencies and mode shapes of lateral, vertical, torsional, longitudinal, and coupled vibrations of the bridge. The results show that the natural frequencies of the bridge are very closely spaced; the first 40 natural frequencies range from 0.068 to 0.616 Hz only. The computed normal modes indicate interactions between the main span and side span, and between the deck, cables, and towers. Significant coupling between torsional and lateral modes is also observed. The numerical results are in excellent agreement with the measured first 18 natural frequencies and mode shapes. The established dynamic model and computed dynamic characteristics can serve further studies on a long-term monitoring system and aerodynamic analysis of the bridge.     

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.     

Cross Correlations of Modal Responses of Tall Buildings in Wind-Induced Lateral-Torsional Motion

Recent trends towards developing increasingly taller and irregularly-shaped buildings imply that these complex structures are potentially more responsive to wind excitation. Making accurate predictions of wind loads and their effects on such structures is therefore a necessary step in the design synthesis process. This paper presents a framework for dynamic analysis of the wind-induced lateral-torsional response of tall buildings with three-dimensional (3D) mode shapes. The cross correlation reflecting the statistical coupling among modal responses under spatiotemporally varying dynamic wind excitations has been investigated in detail. The effects of intermodal correlations on the lateral-torsional response of tall buildings with 3D mode shapes and closely spaced natural frequencies are elucidated and a more accurate method for quantifying intermodal cross correlations is analytically developed. Utilizing the wind tunnel derived synchronous multipressure measurements, a full-scale 60-story asymmetric building of mixed steel and concrete construction is used to illustrate the proposed framework for the coupled dynamic analysis and highlight the intermodal correlation of modal responses on the accurate prediction of coupled building acceleration.     

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.     

Feasibility of a 1,400 m Span Steel Cable-Stayed Bridge

This paper describes the feasibility of 1,400 m steel cable-stayed bridges from both structural and economic viewpoints. Because the weight of a steel girder strongly affects the total cost of the bridge, the writers present a procedure to obtain a minimum weight for a girder that ensures safety against static and dynamic instabilities. For static instability, elastoplastic, finite-displacement analysis under in-plane load and elastic, finite-displacement analysis under displacement-dependent wind load are conducted; for dynamic instability, multimodal flutter analysis is carried out. It is shown that static critical wind velocity of lateral torsional buckling governs the dimension of the girder. Finally, the writers briefly compare a cable-stayed bridge with suspension bridge alternatives.     

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.     

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.     

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.     

Implications of Design Assumptions on Capacity Estimates and Demand Predictions of Multispan Curved Bridges

The paper presents a detailed seismic performance assessment of a complex bridge designed as a reference application of modern codes for the Federal Highway Administration. The assessment utilizes state-of-the-art assessment tools and response metrics. The impact of design assumptions on the capacity estimates and demand predictions of the multispan curved bridge is investigated. The level of attention to detail is significantly higher than can be achieved in a mass parametric study of a population of bridges. The objective of in-depth assessment is achieved through investigation of the bridge using two models. The first represents the bridge as designed (including features assumed in the design process) while the second represents the bridge as built (actual expected characteristics). Three-dimensional detailed dynamic response simulations of the investigated bridge, including soil-structure interaction, are undertaken. The behavior of the as-designed bridge is investigated using two different analytical platforms for elastic and inelastic analysis, for the purposes of verification. A third idealization is adopted to investigate the as-built bridge’s behavior by realistically modeling bridge bearings, structural gaps, and materials. A comprehensive list of local and global, action and deformation performance indicators, including bearing slippage and inter-segment collision, are selected to monitor the response to earthquake ground motion. The comparative study has indicated that the lateral capacity and dynamic characteristics of the as-designed bridge are significantly different from the as-built bridge’s behavior. The potential of pushover analysis in identifying structural deficiencies, estimating capacities, and providing insight into the pertinent limit state criteria is demonstrated. Comparison of seismic demand with available capacity shows that seemingly conservative design assumptions, such as ignoring friction at the bearings, may lead to an erroneous and potentially nonconservative response expectation. The recommendations assist be given to design engineers seeking to achieve realistic predictions of seismic behavior and thus contribute to uncertainty reduction in the ensuing design.     

Mode Shape Corrections for Wind Load Effects

Traditionally, wind analysis procedures based on the “gust loading factor” approach and experimental techniques involving the high frequency base balance and the “stick-type” aeroelastic model test have assumed ideal structural mode shapes, i.e., linear lateral modes and uniform torsional modes. The influence of nonideal mode shapes manifests itself through modifications in the generalized wind load, the structural displacement, the equivalent static wind load (ESWL), and the attendant influence function. This has led to the introduction of several correction procedures, each focusing on an individual feature of the overall response analysis framework. This paper presents a systematic development of correction procedures in terms of correction factors (CFs) to account for nonideal mode shapes in the formulation of generalized load, analysis of structural response, and the derivation of the ESWL. A parameter study is conducted to examine the significance of CFs in estimating various load effects. It is observed that the influence of a nonideal mode shape is actually negligible for the displacement response and the base bending moment, but not so for other load effects, e.g., the base shear and the generalized wind load. Although the existing procedures are effective in correcting the intended response component, they should not be used indiscriminately for other load effects. This paper also presents a correction procedure for the influence of mode shapes on the ESWLs, a loading format that is very attractive for implementation in codes and standards and design practice as well as for the correct interpretation of wind tunnel measurements.     

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.     

Dynamic Analysis of Curved Continuous Multiple-Box Girder Bridges

The use of horizontally curved composite multiple-box girder bridges in modern highway systems is quite suitable in resisting torsional and warping effects induced by highway curvatures. Bridge users react adversely to vibrations of a bridge and especially where torsional modes dominate. In this paper, continuous curved composite multiple-box girder bridges are analyzed, using the finite-element method, to evaluate their natural frequencies and mode shapes. Experimental tests are conducted on two continuous twin-box girder bridge models of different curvatures to verify and substantiate the finite-element model. Empirical expressions are deduced from these results to evaluate the fundamental frequency for such bridges. The parameters considered herein are the span length, number of lanes, number of boxes, span-to-radius of curvature ratio, span-to-depth ratio, end-diaphragm thickness, number of cross bracings, and number of spans.     

Considerations of Multimode Structural Response for Near-Field Earthquakes

This paper presents an investigation of multimode effects of tall buildings idealized as a continuous shear-beam model subjected to near-field pulse-like ground motion. The investigation is based on three analytical approaches: a damped wave solution approach, a fundamental-mode approach, and a modal summation approach. In the modal summation approach, all modal damping ratios are assumed to be equal and a set of Green’s functions for the shear strain response is explicitly derived. The multimode effects on the base-level shear strain/force demands are compared by using an effective response spectrum for shear-beam systems. The study results show that the occurrence of major spectral differences is conditioned on the ratio of the fundamental structural period to the duration of the predominant excitation pulse. Seismic analyses for a set of recorded near-field earthquake data indicate a strong correlation between the characteristics of effective response spectra and the ground pulse parameters.     

Investigation of Turbulence Effects on Torsional Divergence of Long-Span Bridges by Using Dynamic Finite-Element Method

Aerostatic stability of super long-span bridges is a much concerned issue during the design stage. Typical aerostatic instability is the so-called torsional divergence which may lead to abrupt structural failure. The iterative static-based FEM, which generally entails the assumption of smooth oncoming flow, has been widely used to evaluate the aerostatic stability of the bridge concerned. However, the wind in atmospheric boundary layer is naturally turbulent and the effect of turbulence on bridge torsional divergence should be therefore considered, and that is the main concern of the present study. To take into account the effects of turbulence on torsional divergence, a dynamic-based time domain finite-element (FE) procedure for predicting bridge aerostatic stability is introduced first. Then the quasi-steady wind loads expressions are presented and discussed, into which the aerodynamic torsional stiffness, which is indispensable for the evaluation of aerostatic stability, has been demonstrated to be incorporated indirectly by a frequency-domain-based approach. Finally, the aerostatic performances of the longest suspension bridge in China are investigated, of which the torsional divergence is the primary concern. Numerical results show that the torsional divergence pattern in turbulent flow differs considerably from that in smooth flow. The primary difference is, while the torsional instability in smooth flow manifests as an abrupt mounting up of the twist deformation of the main girder with the increasing of the wind velocity, that in turbulent flow manifests as an unstable stochastic vibration with large peak values. Another difference is that the wind velocity for divergence in turbulent flow is obviously lower than that in smooth wind and there does not present an obvious wind velocity threshold for divergence, which is distinguished from the torsional divergence in smooth flow characterized by a clear threshold. Based on the presented time domain FE analysis procedure, the influence of turbulence intensity and gusts spatial correlation upon torsional divergence is also investigated and shown to play an important role on the aerostatic stability.