Dynamic Response Of Suspension Bridge Research Articles

Modal Characteristic and Nonlinear Dynamic Response of Suspension Bridge with Lateral Asymmetric Stiffness

Asymmetric stiffness in transverse direction of suspension bridge can be easily induced by many causes during its long-term service. Such phenomenon may cause the coupling effect between vertical and torsional vibrations. A cross-section model of suspension bridge with seven-degree of freedom is proposed, to investigate the asymmetry effect on the dynamic behavior of the system. Corresponding modal analysis is firstly carried out. Results show that the asymmetric stiffness will induce veering phenomenon when natural frequency loci of vertical and torsional modes approach each other. In the veering region, mode hybridization phenomenon can be observed between these two modes. In addition, asymmetry-induced nonlinear vibration of hybrid vertical and torsional modes is studied using the extended incremental harmonic balance method. The effect of asymmetry extent is also investigated in this study. Results show that both hybrid modes can be excited by either the vertical or torsional excitation. Moreover, the energy can be transferred between these two modes, because of the nonlinear stiffness introduced by the significant swaying motion of hanger and cable.

Influence of Fluid Viscous Damper on the Dynamic Response of Suspension Bridge under Random Traffic Load

Fluid viscous dampers (FVDs) are widely used in long‐span suspension bridges for earthquake resistance. To analyze efficiently the influences of FVDs on the dynamic response of a suspension bridge under high‐intensity traffic flow, a bridge‐vehicle coupling method optimized by isoparametric mapping and improved binary search in this work was first developed and validated. Afterwards, the traffic flow was simulated on the basis of monitored weigh‐in‐motion data. The dynamic responses of bridge were analyzed by the proposed method under different FVD parameters. Results showed that FVDs could positively affect bridge dynamic response under traffic flow. The maximum accumulative longitudinal girder displacement, longitudinal girder displacement, and longitudinal pylon acceleration decreased substantially, whereas the midspan girder bending moment, pylon bending moment, longitudinal pylon displacement, and suspender force were less affected. The control efficiency of maximum longitudinal girder displacement and accumulative girder displacement reached 33.67% and 57.71%, longitudinal pylon acceleration and girder bending moment reached 31.51% and 7.14%, and the pylon longitudinal displacement, pylon bending moment, and suspender force were less than 3%. The increased damping coefficient and decreased velocity exponent can reduce the bridge dynamic response. However, when the velocity exponent was 0.1, an excessive damping coefficient brought little improvement and may lead to high‐intensity work under traffic flow, which will adversely affect component durability. The benefits of low velocity exponent also reduced when the damping coefficient was high enough, so if the velocity exponent has to be increased, the damping coefficient can be enlarged to fit with the velocity exponent. The installation of FVDs influences dynamic responses of bridge structures in daily operations and this issue warrants investigation. Thus, traffic load should be considered in FVD design because structural responses are perceptibly influenced by FVD parameters.

Investigation of dynamic wind loads on a long-span suspension bridge identified from measured acceleration data

The traditional wind load assessment for long-span bridges relies on assumed models for the wind field and aerodynamic coefficients from wind tunnel tests, which usually introduce some uncertainties. Recent studies have shown that large deviations can exist between the predicted and observed wind-induced dynamic response of suspension bridges. In studies of the dynamical behavior of bridges, inverse force identification methods can therefore be an interesting tool in the assessment of possible uncertainties involved in the modeling of wind loads. This paper presents a novel case study of the identification of the dynamic wind loads on the 1310 ​m long Hardanger bridge, a suspension bridge equipped with a monitoring system for wind and vibrations. The modal wind loads are identified from acceleration data using an algorithm for model-based joint input and state estimation. Several data sets with different wind conditions are presented. The wind loads are studied in the time and frequency domains and are compared to the mean velocity and turbulence characteristics of the wind.

Stability of dynamic response of suspension bridges

The potential occurrence of internal parametric resonance phenomena has been recently indicated as a potential contributory cause of the appearance of critical dynamic states in long-span suspension bridges. At the same time, suspension bridges, in view of their flexibility, are prone to aeroelastic response, such as vortex shedding, torsional divergence and flutter. In this paper, a non-linear dynamic model of a suspension bridge is devised, with the purpose of providing a first attempt toward a unified framework for the study of aeroelastic and internal resonance instabilities. Inspired by the pioneering work of Herrmann and Hauger, the analyses have been based on a linearized formulation that is able to represent the main structural non-linear effects and the coupling given by aerodynamic forces. The results confirm that the interaction between aeroelastic effects and non-linear internal resonance leads to unstable conditions for wind speeds which can be lower than the critical threshold for standard aeroelastic predictions.

Dynamic Analysis of Suspension Bridges and Full Scale Testing

This paper is concerned with the earthquake analysis of suspension bridges, in which the effects of large deflections are taken into account. The first part of the study deals with an iteration scheme for the nonlinear static analysis of suspension bridges by means of tangent stiffness matrices. The concept of tangent stiffness matrix is then introduced in the frequency equation governing the free vibration of the system. At any equilibrium stage, the vibrations are assumed to take place tangent to the curve representing the force-deflection characteristics of the structure. The bridge is idealized as a three dimensional lumped mass system and subjected to three orthogonal components of earthquake ground motion producing horizontal, vertical and torsional oscillations. By this means a realistic appraisal is achieved for torsional response as well as for the other types of vibration. The modal response spectrum technique is applied to evaluate the seismic loading for the combination of these vibrations. Various numerical examples are introduced in order to demonstrate the method of analysis. The procedure described enables the designer to evaluate the nonlinear dynamic response of suspension bridges in a systematic manner.

The influence of the higher order modes on the dynamic response of suspension bridges

The influence of the higher order modes of vibrations on the dynamic response of a suspension bridge is studied for two cases of loading. The first case is when the bridge is subjected to wind loading on the suspended cables and the bridge deck. The second case is when the bridge is subjected to a moving load on the bridge deck. The dynamic responses of the bridge deck and the suspended cables considering some higher order modes of vibrations are compared with the bridge response when it is modeled by the first dominant modes for each case of loading. It is shown that the influence of the higher order modes when the suspension bridge is subjected to wind loading is more significant than in the case when the bridge is subjected to a moving load. Therefore, it is recommended to investigate the influence of some higher order modes on the dynamic response of the suspension bridges and not always to depend on the response estimated from the first dominant modes of vibration.

Time domain modeling of self-excited aerodynamic forces for cable-supported bridges: A comparative study

Prediction of the wind-induced dynamic response of suspension bridges, emphasizing self-excited forces, is discussed in this paper. The self-excited forces have been modeled by two commonly applied unsteady models and an unsteady model introduced and explained in this article. A novel frequency-independent approximation of the self-excited forces, which for the suspension bridge considered provides results as accurate as those from the unsteady models, is also presented. An integration method that may reduce the number of time steps necessary to avoid amplitude and phase distortion of the self-excited forces has been introduced and applied successfully in a comprehensive case study.

Effect of Hanger Flexibility on Dynamic Response of Suspension Bridges

Linearized continuum models of a suspended span with unloaded backstays and of a symmetric three-span suspension bridge are used to study the effects of the flexibility of the hangers on the vertical vibrations of suspension bridges. The models include elastic parabolic cables, flexible distributed hangers with variable length, and a stiffening girder represented by an elastic beam. It is shown that the free vibrations of a suspended span with unloaded backstays are controlled by five dimensionless parameters, while six dimensionless parameters control the response of a symmetric three-span suspension bridge. The results indicate that the flexibility of the hangers has a significant effect on the natural frequencies of the higher modes only when the relative stiffness of the girder is very high. The effects of hanger flexibility on the response of a suspension bridge to localized impulsive loads are also found to be small. These findings confirm the traditional, albeit previously untested, assumption of inextensible hangers. Finally, the threshold amplitudes of free vibrations that would result in the incipient slackening of the hangers are determined.

Full-scale measurements of dynamic response of suspension bridge subjected to environmental loads using GPS technology

Nowadays, there are many larger and taller engineering structures than in the past. These structures are commonly designed to be more flexible. Thus, they are sensitive to severe wind gust and earthquake tremor. For the dynamic response prediction, finite element (FE) modal updating and structural health monitoring, the dynamic characteristics of a structure are becoming increasingly important. The aim of this paper is to show an emerging tool-the real time kinematic (RTK) global positioning system (GPS), which is used to measure and monitor the low-frequency vibration of Dalian BeiDa Bridge, a medium span suspension bridge. The modal analysis is firstly performed on the developed three-dimensional FE model according to the original blue prints of the bridge to provide the analytical frequencies and mode shapes. The field ambient vibration tests using the accelerometers on the bridge deck are conducted just prior to the use of the GPS system in order to validate the FE model. And then, the calibration test is done to assess the dynamic displacement measurement accuracy of the GPS in three orthogonal directions. The GPS signal properties, such as the amplitude, standard deviations (STD), power spectral density (PSD), and probability density functions (PDF), are studied in details. After that, a structural health monitoring system based on the GPS is devised and installed on the bridge to provide real-time measurement of deck movement. The output-only modal parameter identification is carried out by using the Periodogram method to investigate the responses of the bridge. The result shows that the spectral densities of the displacements measured by the GPS correspond well with the results by the FEM analysis and the accelerometer vibration test. It is concluded that the GPS method is reliable and useful to clarify the ambient vibration response behaviors of the flexible structures. The integration of the GPS and accelerometer methods results in a hybrid arrangement that can help to eliminate some disadvantages of the two devices used separately.

Dynamic Response of Suspension Bridge to Typhoon and Trains. I: Field Measurement Results

A wind and structural health monitoring system (WASHMS) was installed in the Tsing Ma suspension bridge in Hong Kong in 1997. On September 16, 1999, Typhoon York, which was the strongest typhoon since 1983 and the typhoon of the longest duration on record, crossed over Hong Kong. All vehicles, except trains, were prohibited from running on the bridge for 2.5h. This event provided a distinctive opportunity to examine the existing analytical models for predicting dynamic response of long suspension bridges to high winds and running trains. The field measurement data recorded by the WASHMS during this event were therefore analyzed and the four particular cases were identified and discussed in this paper. The four particular cases identified included: (1) the bridge without any vehicles; (2) the bridge with one train; (3) the bridge with two trains running in opposite directions; and (4) the bridge with three running trains. The mean wind speed was almost perpendicular to the bridge alignment in the four cases to facilitate the comparison with analytical results. For each case, wind characteristics, bridge acceleration responses, and bridge displacement responses were analyzed using the measurement data from anemometers, accelerometers, and level sensing systems, respectively. The number of trains running on the bridge, train speed, and train location were also identified using the measurement data from strain gauges. The measurement results clearly demonstrated the dynamic behavior of the bridge with running trains during high winds. The measurement results obtained will also be used to verify the analytical model developed by the authors in the companion paper.

Dynamic Response of Suspension Bridge to Typhoon and Trains. II: Numerical Results

The field measurement results of the Tsing Ma suspension bridge under cross winds and running trains during Typhoon York have been presented in Part I of this study. The field measurement results are used in this paper (Paper II) to verify, to some extent, the finite element based framework developed by the writers in the time domain for predicting dynamic response of coupled train and bridge systems in cross winds. In this regard, a three-dimensional finite element model is used to model the Tsing Ma suspension bridge. Wind forces acting on the bridge are generated in the time domain using a fast spectral representation method and based on the field measurement results and wind tunnel test results. Each coach in a train running on the bridge is modeled by a 27 degree-of-freedom dynamic system. The dynamic interaction between the bridge and the train is realized through the contact forces between the wheels and the track. Track irregularity is simulated using a high-order multivariate autoregressive model, whereas wheel hunting is assumed to be a sinusoidal function with a random phase. The acceleration and displacement responses of the bridge under cross winds and running trains are computed and compared with the responses measured from the field for four cases. The comparison is found to be satisfactory in general. The dynamic response and performance of trains on the bridge is also computed and evaluated. The proposed framework is found to be feasible and practical.

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.

PARAMETRIC ANALYSES OF NATURAL FREQUENCIES OF LOWER VIBRATION MODES OF SUSPENSION BRIDGES

It is well known that lower modes of vibration are predominant in the dynamic response of suspension bridges to moving vehicles or wind. Therefore, it is needed to know their characterisics, in particular the corresponding natural frequencies.In this brief paper, natural frequencies of lower vibration modes of suspension bridges are numerically examined, mainly with respect to a side to center span ratio besides nondimensional characteristic parameters introduced in the previous paper 2).It is found from these results that the characteristics of symmetric natural frequencies of suspension bridges associated with lower modes depend significantly on the values of the side to center span ratio.

Finite element analysis of suspension bridges

A finite element analysis of static and dynamic response of suspension bridges is presented in this paper. Several finite element models of the three-dimensional bridge structure, with varying degrees of complexity and accuracy, are discussed. The formulation takes into account the geometric nonlinearities of the cables and some elements of the girders-bracingsdeck system as well as the nonlinear material properties of the components. Special attention is given to the effects of steady and unsteady wind forces. Examples of application include calculations of the static and dynamic response of a bridge subjected to wind and moving loads.