Self-excited Forces Research Articles

Prediction of long-term extreme load effects due to wind for cable-supported bridges using time-domain simulations

Many long-span bridges that are slender and susceptible to wind-induced vibrations are currently under construction all over the world. As nonlinear structural behavior and nonlinear displacement-dependent wind loads become of higher importance, time-domain methods are commonly applied instead of frequency-domain approaches. This paper discusses how rational functions, fitted to aerodynamic derivatives, can be converted to a state-space model to transform the frequency-dependent aerodynamic forces into the time domain. A user element has been implemented in ABAQUS to include the self-excited forces in the dynamic analysis. The flutter stability limit and buffeting response of the Hardanger Bridge have been calculated in a comprehensive case study to illustrate the performance of the presented methodology. The Average Conditional Exceedance Rate method is used to estimate long-term extreme load effects. A simplified full long-term method that ignores wind velocities which contribute little to the long-term extreme values is introduced to reduce the required computational effort. The long-term predictions are also compared with results obtained using a short-term approach and it is concluded that the long-term extreme load effects are approximately 14% higher than the short-term extreme values.

The Hardanger Bridge monitoring project: Long-term monitoring results and implications on bridge design

In design of slender suspension bridges, which are prone to wind excitation, accurate prediction of dynamic response is essential for reliable designs. However, dynamic response calculations involve many sources of uncertainty, one of which is the modeling of the gust loading. The long-term data of wind velocities and accelerations from the Hardanger Bridge monitoring project are presented here to investigate the variability in dynamic response. The buffeting response of the bridge is then evaluated in frequency domain using a multimode approach. The self-excited forces are modeled using aerodynamic derivatives obtained from free vibration tests. The modal properties of the bridge are extracted from a finite element model. The spectral densities and coherences required to describe the gust loading are modeled using design provisions. The analytical results are compared with the full-scale measurement results and the implications on design of long-span suspension bridges are discussed.

Influence of visual information on pedestrian actions on laterally oscillating structures – experimental study using virtual reality environments

A new line of inquiry into the causes of lateral instability of structures due to actions of walking pedestrians has been opened recently thanks to the fusion of structural dynamics, biomechanics and experimental psychology. This has been achieved by means of a novel experimental setup. The setup consists of an instrumented treadmill, supported by a shaking table providing lateral motion, a motion capture system for monitoring kinematics of walkers, a treadmill speed feedback control mechanism for automatic adjustment of the treadmill speed to that of the walker and immersive virtual reality implemented to provide similarity of the visual environment with real life experience. It has been previously observed that the visual environment can influence pedestrian behaviour and the resulting ground reaction forces. Statistically significant differences between gait parameters and self-excited forces, critical for structural stability, have been measured during tests in which the visual environment was either that of a laboratory, containing an abundance of stationary visual reference cues, or virtual reality representative of walking on a footbridge. This highlighted the importance of providing visual information representative of real life experience. Another opportunity presented by the experimental setup is explored in this study in that it allows the influence of differently structured visual environments to be tested. To this end, two types of visual scenarios are implemented in virtual reality, differentiated by the amount of visual information available to the walker. It is shown that the visual information can indeed alter pedestrian behaviour and that reducing the amount of visual information can have a detrimental effect for structural stability. This raises a question whether a number of design recommendations proposed in recent years, calibrated based on data from laboratory-based investigations neglecting the influence of visual information on pedestrian behaviour, should be reconsidered and updated.

Influence of Low-Frequency Vertical Vibration on Walking Locomotion

Walking locomotion has been a subject of studies in diverse research fields, such as computer, medical, and sport sciences, biomechanics, and robotics, resulting in improved understanding of underlying body motion and gait efficiency and pathology (when present). Only recently, a detailed understanding of kinematics and kinetics of the walking locomotion has become an important requirement in structural engineering applications due to an increasing sensitivity of modern, lightweight, low-frequency, and lightly damped footbridges to pedestrian-induced dynamic excitation. To facilitate development, calibration and verification of pedestrian models requires experimental characterization of walking gait parameters and understanding whether and how these parameters are influenced by the structural vibration. This study investigates whether low-frequency vibrations in the vertical direction affect seven walking locomotion parameters: pacing frequency, step length, step width, angle of attack, end-of-step angle, trunk angle, and amplitude of the first forcing harmonic. Three participants took part in a testing program consisting of walking on a treadmill placed on both stationary and vibrating supporting surfaces. The collected data suggest that an increasing level of vibration results in an increase in step-by-step variability for the majority of parameters. Furthermore, the existence of the self-excited force, previously observed only in numerical simulations of walking on pre-excited bridge decks, was confirmed. In addition, the deck vibration tended to have a beneficial effect of reducing the net force induced into the structure when walking at a pacing rate close to the vibration frequency. Finally, it was found that the vibration level perceptible by a pedestrian is one to two orders of magnitude larger than that typical of a standing person, and that the sensitivity to vibration decreases as the speed of walking increases.

Analysis of self-excited forces for a box-girder bridge deck through unsteady RANS simulations

This paper presents the results of two-dimensional numerical simulations of the flow field around a trapezoidal box-girder bridge section with later cantilevers, experiencing small-amplitude heaving or pitching harmonic oscillations. Unsteady Reynolds-averaged Navier–Stokes equations are solved in conjunction with an eddy-viscosity and an explicit algebraic Reynolds stress model. Flutter derivatives are determined and compared with wind tunnel results, showing fairly good agreement. The degree of sharpness of the deck lower edges is found to play a key role in the aeroelastic behavior of the profile. In particular, the bridge section fully behaves as a bluff body and is prone to low-reduced-wind-speed torsional galloping in the case of perfectly sharp edges. By contrast, the presence of a small radius of curvature in the section lower corners changes the nature of the instability to coupled flutter and significantly postpones the stability threshold, in line with a quasi-streamlined body behavior. Moreover, a wide sensitivity study is carried out, investigating the influence on the self-excited forces of the amplitude of oscillation, mean angle of attack and Reynolds number. In particular, the numerical simulations for the geometry with smooth lower edges highlight the regime transition occurring when the Reynolds number is varied, with significant effects on the flutter derivatives. Finally, the numerical flow visualizations provide a physical explanation of some phenomena observed in the wind tunnel experiments.

Effects of Exponentially Modified Sinusoidal Oscillation and Amplitude on Bridge Deck Flutter Derivatives

With the use of a computational fluid dynamics (CFD) numerical simulation, in this study an unusual exponentially modified sinusoidal (EMS) oscillation was forcibly imposed on deck section to simulate amplitude-varying self-excited forces. This approach facilitated the investigation of the influence of EMS transient oscillation and amplitude on self-excited forces and flutter derivatives. Study results show that for an ideal thin plate, the sensitivities of H2*, H4*, A2*, and A4* to oscillation decaying/diverging ratio (ξ) were higher than those of H1*, H3*, A1*, and A3*. With different amplitudes, ξ, and reduced wind velocities, the self-excited forces of one flat plate and two bluff bridge decks were numerically calculated and the flutter derivatives were extracted and compared with the theoretical and experimental results. If |ξ| < 0.1, its influence was almost negligible, by which the traditional sinusoidal oscillation technique was validated. The heaving and torsional amplitudes had different influences on each flutter derivative. The influences were also related to deck cross sections and reduced wind velocities. The proposed numerical simulation of EMS oscillation enables a better understanding of the influence of oscillation amplitude and ξ on bridge deck flutter derivatives.

Investigation of Influence Factors of Wind-Induced Buffeting Response of a Six-Tower Cable-Stayed Bridge

This paper presents an investigation of the wind-induced buffeting responses of the Jiashao Bridge, the longest multispan cable-stayed bridge in the world. A three-dimensional finite element model for the Jiashao Bridge is established using the commercial software package ANSYS and a 3D fluctuating wind field is simulated for both bridge deck and towers. A time-domain procedure for analyzing buffeting responses of the bridge is implemented in ANSYS with the aeroelastic effect included. The characteristics of buffeting responses of the six-tower cable-stayed bridge are studied in some detail, focusing on the effects including the difference in the longitudinal stiffness between the side towers and central towers, partially longitudinal constraints between the bridge deck and part of bridge towers, self-excited aerodynamic forces, and the rigid hinge installed in the middle of the bridge deck. The analytical results can provide valuable references for wind-resistant design of multispan cable-stayed bridges in the future.

Time Domain Flutter Speed Analysis of Cable Stayed Bridge

Modelling of cable stayed bridges subjected to wind loading is considered in this paper. A geometrically nonlinear static analysis is performed, based on which cable pretensions and system mass and stiffness matrices are obtained. The extended Scanlan-Tomko relations that include vertical, torsional, and lateral degrees of freedom are considered for the self-excited forces. The rational function approach is used for obtaining flutter derivatives from available experimental data. A time domain approach is adopted, which yields modified mass, stiffness and damping modal matrices and also augmented (aerodynamic) states. The modal state-space equations are thus obtained, and the eigenvalue problem solved to find the flutter speed and mode.

Higher-Order Self-Excited Drag Forces on Bridge Decks

AbstractNonlinear features concerning self-excited drag forces induced by the vertical and torsional motions for typical deck sections are investigated comprehensively in this study. The self-excited drag forces are calculated using computational fluid dynamics (CFD)-based numerical simulations. In a conventional analysis framework, the self-excited drag force is modeled as a linear function of the structural motions. However, the simulation results from CFD indicate that in many instances, the second-order (nonlinear) component of the self-excited drag force is more significant than the first-order (linear) component. To enhance the modeling fidelity of a conventional aeroelastic analysis framework on the basis of the semiempirical flutter derivative concept, a nonlinear mathematical model for characterizing both the first- and second-order components is developed to better quantify the self-excited forces and more accurately extract the flutter derivatives. Its efficacy and superiority compared with the...

Framework for sensitivity and uncertainty quantification in the flutter assessment of bridges

The phenomenon of aerodynamic instability caused by wind is usually a major design criterion for long-span cable-supported bridges. If the wind speed exceeds the critical flutter speed of the bridge, this constitutes an Ultimate Limit State. The prediction of the flutter boundary therefore requires accurate and robust models. The state-of-the-art theory concerning determination of the flutter stability limit is presented. Usually bridge decks are bluff and therefore the aeroelastic forces under wind action have to be experimentally evaluated in wind tunnels or numerically computed through Computational Fluid Dynamics (CFD) simulations. The self-excited forces are modelled using aerodynamic derivatives obtained through CFD forced vibration simulations on a section model. The two-degree-of-freedom flutter limit is computed by solving the Eigenvalue problem.A probabilistic flutter analysis utilizing a meta-modelling technique is used to evaluate the effect of parameter uncertainty. A bridge section is numerically modelled in the CFD simulations. Here flutter derivatives are considered as random variables. A methodology for carrying out sensitivity analysis of the flutter phenomenon is developed. The sensitivity with respect to the uncertainty of flutter derivatives and structural parameters is considered by taking into account the probability distribution of the flutter limit. A significant influence on the flutter limit is found by including uncertainties of the flutter derivatives due to different interpretations of scatter in the CFD simulations. The results indicate that the proposed probabilistic flutter analysis provides extended information concerning the accuracy in the prediction of flutter limits.The final aim is to set up a method to estimate the flutter limit with probabilistic input parameters. Such a tool could be useful for bridge engineers at early design stages. This study shows the difficulties in this regard which have to be overcome but also highlights some interesting and promising results.

Imperfect Correlation of Vortex-Induced Fluctuating Pressures and Vertical Forces on a Typical Flat Closed Box Deck

A wind tunnel test of synchronized pressure measurement on a spring-suspended sectional model of a flat closed box deck at a large scale of 1/20 was carried out by taking Xiangshan Harbour Bridge in China as background. The chordwise correlation of the fluctuating aerodynamic pressures (FAP) on the surface of the box deck was then investigated at both stationary state and vortex-induced resonance (VIR) states of the deck model. Afterwards, an approximate approach was developed for estimating the spanwise correlation coefficients of pure vertical vortex-shedding force (VSF) on the deck under VIR state. On this basis, the spanwise correlation coefficients of both the integrated total vertical fluctuating aerodynamic force (FAF) and the pure vertical VSF on the flat box deck were then studied and compared to each other quantitatively at the stationary state as well as the VIR states at various wind speeds. Finally, the spanwise correlation coefficients of the surface FAP at some typical positions on the deck cross section were calculated and compared with that of the total vertical FAF and the vertical VSF. The results show that the chordwise correlation of the FAP on stationary deck is not strong at most part of the deck surface, however, can be enhanced with the increase of incident wind speed. The chordwise correlation of FAP can also be enhanced quite significantly by the deck oscillation under VIR states, but is still imperfect. Furthermore, the spanwise correlations of FAP at different positions on the deck cross section are clearly different from each other, and much weaker than that of the total vertical FAF, for both the stationary state and the VIR states of the deck. However, they are generally much stronger than that of the pure VSF in cases of notable VIR, and much weaker in the cases of slight VIR, such as near the onset of VIR. The spanwise correlation characteristics of both the pure vertical VSF and the total vertical FAF are hence very different from those of the surface pressures. Moreover, the spanwise correlation of the total vertical FAF on cross sections can be greatly intensified by the VIR because of the existence of a large portion of vortex-induced self-excited force (VISEF). On the other hand, the spanwise correlation of the vertical VSF under the VIR states is much weaker than that of total vertical FAF as well as that of the pure vertical VSF under the stationary state at same wind speeds. Finally, the variations of the spanwise correlation of both the total vertical FAF and the pure vertical VSF with the response displacement of VIR is quite complicated. The most violent VIR, the most correlated total FAF and the most correlated VSF occur generally at different wind speeds.

Experimental identification of the behaviour of and lateral forces from freely-walking pedestrians on laterally oscillating structures in a virtual reality environment

Modelling pedestrian loading on lively structures such as bridges remains a challenge. This is because pedestrians have the capacity to interact with vibrating structures which can lead to amplification of the structural response. Current design guidelines are often inaccurate and limiting as they do not sufficiently acknowledge this effect. This originates in scarcity of data on pedestrian behaviour on vibrating ground and uncertainty as to the accuracy of results from previous experimental campaigns aiming to quantify pedestrian behaviour in this case. To this end, this paper presents a novel experimental setup developed to evaluate pedestrian actions on laterally oscillating ground in the laboratory environment while avoiding the implications of artificiality and allowing for unconstrained gait. A biologically-inspired approach was adopted in its development, relying on appreciation of operational complexities of biological systems, in particular their adaptability and control requirements. In determination of pedestrian forces to the structure consideration was given to signal processing issues which have been neglected in past studies. The results from tests conducted on the setup are related to results from previous experimental investigations and outputs of the inverted pendulum pedestrian model for walking on laterally oscillating ground, which is capable of generating self-excited forces.

Applicability of URANS and DES Simulations of Flow Past Rectangular Cylinders and Bridge Sections

This paper discusses the results of computational fluid dynamics simulations carried out for rectangular cylinders with various side ratios of interest for many civil engineering structures. A bridge deck of common cross-section geometry was also considered. Unsteady Reynolds-averaged Navier–Stokes (URANS) equations were solved in conjunction with either an eddy viscosity or a linearized explicit algebraic Reynolds stress model. The analysis showed that for the case studies considered, the 2D URANS approach was able to give reasonable results if coupled with an advanced turbulence model and a suitable computational mesh. The simulations even reproduced, at least qualitatively, complex phenomena observed in the wind tunnel, such as Reynolds number effects for a sharp-edged geometry. The study focused both on stationary and harmonically oscillating bodies. For the latter, self-excited forces and flutter derivatives were calculated and compared to experimental data. In the particular case of a benchmark rectangular 5:1 cylinder, 3D detached eddy simulations were also carried out, highlighting the improvement in the accuracy of the results with respect to both 2D and 3D URANS calculations. All of the computations were performed with the Tau code, a non-commercial unstructured solver developed by the German Aerospace Center.

Coupled flutter analysis of long-span bridges using full set of flutter derivatives

A convenient and effective finite element-based method for coupled flutter analysis of long-span bridges is presented. The exact formulation of the aerodynamic self-excited forces with eighteen flutter derivatives utilized by complex notation is proposed. The predictions of the flutter wind speed and the critical frequency are compared with those either given by existing methods or the wind tunnel test showing the effectiveness and accuracy of the present approach. Numerical flutter analysis for an asymmetric bridge is the application for engineering practice, and its obtained results highlight the important role of the first lateral bending and torsional mode in generating the coupled flutter. Multi-mode analyses that are based on only the symmetrical modes can predict accurately the bridge flutter onset. The consistent self-excited aerodynamic force formulations produce the flutter velocity that is closer to the experimental one of full-bridge model in the wind tunnel.

Nonlinearity of mechanical damping and stiffness of a spring-suspended sectional model system for wind tunnel tests

The wind tunnel test of spring-suspended sectional models (SSSM) is an important means in the research of wind engineering, which is very frequently employed to check the performances of flutter and vortex-induced resonance of bridges as well as to identify the various aerodynamic and aeroelastic parameters of bridge components, such as aerodynamic derivatives of self-excited forces. However, in practice, the mechanical damping ratios and natural frequencies of SSSM system are prevailingly supposed to be constant in the whole procedure of a test. This assumption often leads to notable errors of the test results or dispersion of the identified aerodynamic parameters because the mechanical damping ratios and natural frequencies of SSSM system are proved to vary in fact to some extent with the change of oscillating amplitude. On that account, the mechanical nonlinearity of SSSM system is investigated and discussed in this paper by taking a flat-closed box section as a research background. The conventional linear model is firstly proved to fail to predict precisely the long-duration free decay responses of the SSSM system. The formulae of equivalent linearization approximation (ELA) are then derived by using a multiple-scale method to model the mechanical nonlinearities in the first-order approximate sense, and a time-domain system identification method is proposed on this basis to identify equivalent amplitude-dependent (EAD) damping ratio and frequency. The proposed ELA and nonlinear system identification methods are then found to be precise enough to model the mechanical nonlinearities of SSSM system. The characteristics of EAD damping ratio and frequency of both the bending and torsional modes are then discussed in detail. It is then found that the major energy dissipation of SSSM vibrations at both the bending and torsional modes generally comes from the combined effect of viscous damping and quadratic damping. However, for the vibration at the bending mode with small to moderate amplitudes, the coulomb friction damping becomes the major source of energy dissipation. Furthermore, different types of additional dampers added on the SSSM system will introduce extra mechanical nonlinearities in manners very different from each other. The liquid viscous damper with oil tank form employed in this study is found to be an ideal linear damper, whereas the steel wire-rope-circle damper of friction type brings in significant mechanical nonlinearities.

A simulation study on the optimal control of buffeting displacement for the Sutong Bridge with multiple tuned mass dampers

The buffeting of long-span cable-supported bridges under strong winds is one of the key issues in bridge wind engineering. In order to study the effectiveness of the multiple tuned mass dampers (MTMDs) in buffeting control of long-span bridges, the Sutong Cable-stayed Bridge (SCB) with a main span of 1088 m in China is taken as an example in this paper. The spatial finite element model of the SCB is established and the modal analysis is conducted based on ANSYS. After the 3D turbulence wind field of the SCB is simulated using the measured wind parameters, the time-domain buffeting analysis on the SCB is conducted with the aerodynamic self-excited forces included. According to the dynamic characteristics and the time-domain buffeting analysis results of the SCB, the parameter sensitivity analysis on buffeting vibration control with MTMD is conducted in ANSYS. The optimum parameters are then obtained with the construction difficulty and economic factors considered. Results show that the control efficiency is sensitive to the number of the TMD, mass ratio, frequency band-width ratio, and damping ratio. Both the vertical and the lateral vibrations can be effectively controlled when proper design parameters of a MTMD system are used. In addition, the control effect on lateral vibration is better than that on vertical vibration. Results obtained in this study can provide references for anti-wind design and buffeting control of long-span cable-stayed bridges.

Experimental study on aerodynamic derivatives of a bridge cross-section under different traffic flows

The presence of traffic on a bridge deck has an important primary impact on the bridge cross-section׳s aerodynamic configuration, which may affect the static force coefficients and the aerodynamic derivatives that will in turn influence the self-excited forces acting on the bridge. Compared to the investigations of the traffic effect on the static force coefficients, its effect on the aerodynamic derivatives is rather rarely studied. In the present study, the forced vibration device is adopted in the wind tunnel to test a scaled bridge section model with vehicle models distributed on the bridge deck considering three simulated traffic flows. The influence of vehicles of different traffic flows under different wind attack angles on the aerodynamic derivatives of the bridge with the reduced wind velocity are investigated based on the experimental results. The measured results show that the influence of vehicles on the aerodynamic derivatives of the bridge cannot be ignored. The impacts of vehicles under different traffic flows on the aerodynamic derivatives are different for different aerodynamic derivatives. The effect of vehicles on the aerodynamic is different for different wind attack angles and different for different traffic flow conditions at a certain attack angle.

The influence of self-excited forces on wind loads and wind effects for super-large cooling towers

The self-excited force induced by the interaction between the structure and airflow has been technically ignored in previous studies conducted on wind loads and wind effects for super-large cooling towers. Moreover, no criteria could be used for qualitative and quantitative studies of the effects resulting from the self-excited force. In this study, by taking a Chinese super-large cooling tower for nuclear power with a height of 215m as an example, the wind tunnel test of cooling tower aero-elastic model with simultaneous pressure and vibration measurement through a modified equivalent beam-net design method was firstly carried out, in which external wind loads of super-large cooling towers were obtained with the presence of the self-excited force. Firstly, the effects of self-excited force on distribution characteristics of external mean and fluctuating wind pressures were analyzed. Through the refined frequency-domain algorithm of wind-induced responses from our previous studies, wind-induced responses of super-large cooling towers were calculated by considering two cases of wind tunnel tests with or without self-excited force. Subsequently, the effects of self-excited force on wind-induced responses and Gust Response factors of super-large cooling towers were discussed. The results showed that the self-excited force had weak influence on the distribution of external mean wind pressure of the super-large cooling tower, while it significantly affected distribution and values of fluctuating wind pressure. Moreover, the effects of self-excited force on wind-induced mean response were negligible for the super-large cooling tower. The response differences calculated using two aerodynamic parameters were mostly below 5%, and regardless of whether or not to consider the influence induced by self-excited force. The excitation mode of wind-induced resonant response was consistent even at the maximum error (i.e. 10%). Furthermore, this study indicated that the influence of self-excited force on wind effects was advantageous to majority of regional structures. To overcome the problems existing in aero-elastic model, e.g., accuracy, accessibility and cost of test, this study suggested that external wind load model without self-excited forces obtained by pressure test of rigid body could be used to calculate the wind-induced responses of super-large cooling towers, in which the differences were within an acceptable range.

Experimental identification of the lateral human–structure interaction mechanism and assessment of the inverted-pendulum biomechanical model

Within the context of crowd-induced lateral bridge vibration, human–structure interaction (HSI) is a widely studied phenomenon. Central to this study is the self-excited component of the ground reaction force (GRF). This force harmonic, induced by a walking pedestrian, resonates with lateral deck motion, irrespective of the pedestrian׳s pacing frequency. Its presence can lead to positive feedback between pedestrian GRFs and structural motion. Characterisation of the self-excited force as equivalent structural mass and damping has greatly improved the understanding of HSI and its role in developing lateral dynamic instability. However, despite this evolving understanding, a key question has remained unanswered; what are the features of a pedestrian׳s balance response to base motion that gives rise to the self-excited force? The majority of the literature has focussed on the effects of HSI with the underlying mechanism receiving comparatively little attention. This paper presents data from experimental testing in which 10 subjects walked individually on a laterally oscillating treadmill. Lateral deck motion as well as the GRFs imposed by the subject was recorded. Three-dimensional motion capture equipment was used to track the position of visual markers mounted on the subject. Thus whole body response to base motion was captured in addition to the GRFs generated. The data presented herein supports the authors’ previous findings that the self-excited force is a frequency sideband harmonic resulting from amplitude modulation of the lateral GRF. The gait behaviour responsible for this amplitude modulation is a periodic modulation of stride width in response to a sinusoidally varying inertia force induced by deck motion. In a separate analysis the validity of the passive inverted pendulum model, stabilised by active control of support placement was confirmed. This was established through comparison of simulated and observed frontal plane CoM motion. Despite the relative simplicity of this biomechanical model, remarkable agreement was observed.

Probabilistic criteria for lateral dynamic stability of bridges under crowd loading

Probabilistic conditions for lateral stability of bridges are proposed, based on output from the inverted pendulum pedestrian model from the field of biomechanics. Statistical variations of the parameters defining the model are studied based on real statistical data of the English population. Variability of the self-excited forces is quantified for crowds of different velocities and critical conditions are identified for bridge natural frequencies below 5Hz. Allowance is made for the influence of the bridge mode shape and number of pedestrians in the crowd and their spatial distribution. This allows realistic worst case conditions among different loading scenarios for a particular structure to be found.