Comparative and sensitivity study of flutter derivatives of selected bridge deck sections,Part 2: Implications on the aerodynamic stability of long-span bridges

This study is the continuation of a comprehensive investigation on section-model aeroelastic coefficients for bridge decks (flutter derivatives) extracted from wind tunnel section-model tests. The original motivation emerged from the United States—Japan Benchmark Study on Bridge Flutter Derivatives, which promoted a series of systematic comparisons of experimental data extracted by two laboratories (Iowa State University, USA and Public Works Research Institute, Japan) as well as previous results available in the literature. Comparisons, which included both streamlined and bluff deck girder models, were summarized in a companion paper [Sarkar P, Caracoglia L, Haan FL, Sato H, Murakoshi J. Comparative and sensitivity study of flutter derivatives of selected bridge deck sections. Part 1: Analysis of inter-laboratory experimental data. Eng Struct 2009;31(1):159–69]. Differences in the flutter derivatives were mainly attributed to: distinct experimental methods in the wind tunnel (free or forced vibration methods), intrinsic variability between different laboratory environments and effects of amplitude dependency in the tests (for bluff sections).In this paper, a sensitivity study was performed to examine the implications of the perceived dissimilarities among flutter-derivative data sets discussed in [Sarkar P, Caracoglia L, Haan FL, Sato H, Murakoshi J. Comparative and sensitivity study of flutter derivatives of selected bridge deck sections. Part 1: Analysis of inter-laboratory experimental data. Eng Struct 2009;31(1):159–69], on the aeroelastic instability of long-span bridges.Numerical analyses were conducted to evaluate flutter instability boundaries of a set of long-span bridge configurations. Both single-mode and coupled-mode instability were considered, depending on the cross section type and characteristics. It is concluded that uncertainty in flutter derivatives occurring as a result of extraction method or intrinsic variability between different laboratories from negligibly small values to as much as fifty percent, as observed in [Sarkar P, Caracoglia L, Haan FL, Sato H, Murakoshi J. Comparative and sensitivity study of flutter derivatives of selected bridge deck sections. Part 1: Analysis of inter-laboratory experimental data. Eng Struct 2009;31(1):159–69], do not affect the variability in the predicted critical velocity in a proportional way. However, differences in the resulting critical velocities have been observed and estimated from as small as five percent to more than thirty percent, heavily depending on the type of bridge, the simulated conditions and type of instability, either dominated by a single mode or influenced by modal coupling.     

桥板颤振导数的比较和敏感性分析 第一部分：试验数据分析

为评估风载下桥板的性能,通常采用多年来发展的两个独特的方法（自由和强迫振动）,从风洞模型试验中得到桥板气动弹性系数。尽管已有众多的研究者对每个技术优缺点进行分析,但文献中缺少对试验结果的系统比较。本研究的意义在于进行与长期颤振导数试验数据相关的评估,包括分析和解释自由和强迫振动两种方法的异同点。从2002年开始,美国的爱荷华州大学和日本的公立研究所着手进行桥颤振导数的研究。试验包括的截面类型很广,从矩形棱柱到改进型,尤其考虑了当前在大跨桥中采用空气动力学设计梁的趋势。同时系统地分析和比较了两种方法颤振导数的试验结果。在相关的论文中,对敏感性进行分析以研究大跨桥在气动弹性不稳定时颤振导数数据中所隐含的不同点。     

Influence of uncertainty in selected aerodynamic and structural parameters on the buffeting response of long-span bridges

Multimode-analysis methods for the study and derivation of flutter instability and buffeting response are readily available from the literature and have been successfully applied to the assessment of the susceptibility of long-span bridges to wind loading. In both cases flutter critical velocity and buffeting oscillation are usually estimated from deterministic analyses. However, the probabilistic nature of the problem is latent since uncertainties, especially those associated with the definition of wind and aerodynamic characteristics, are intrinsically present. These quantities include, for example, wind-turbulence power spectral density, static coefficients and aerodynamic derivatives, usually derived from either site observations or experimental analysis. Their effects are often neglected or usually addressed through sensitivity analyses only.While in the past uncertainty in flutter estimates has been analyzed by researchers (for example through reliability analysis), little attention has been devoted to buffeting. In this paper the effects associated with the random nature of wind and structural characteristics are analyzed through the derivation of a closed-form solution associated with the single-mode buffeting problem with selected random parameters. A specific example is provided to clarify the role of wind power spectral density, damping and selected aeroelastic derivatives.     

紊流风场中桥梁气动导数识别的随机方法

气动导数是大跨桥梁结构颤振和抖振分析的重要依据。本文提出采用随机系统识别方法来识别紊流风场中的气动导数, 与当前应用较广的瞬态激励法及强迫激励法相比, 这类方法的优势在于: (1) 将紊流看作是激励, 而不是噪声, 更能反映结构实际工作状态下的特性; (2) 识别精度不受风速的制约, 可以获得较高折减风速下的气动导数; (3) 可直接在紊流风场中结构随机响应上进行识别, 无需任何人为外在激励, 试验更为简单易行。在风洞中完成了紊流风场中桥梁节段模型测振试验, 进一步利用本文的方法识别出气动导数。与相关文献提供的类似模型在均匀场和紊流场中识别结果的对比表明: 本文识别的气动导数是可靠的, 所提出的采用随机系统识别方法来识别紊流风场中气动导数的思路是可行的。     

风雨共同作用下平板模型的气动导数试验研究

气动导数是大跨桥梁结构颤振和抖振分析中确定颤振临界风速和抖振响应的重要依据。在实际中 ,强风(特别是台风 )一般总伴随着暴雨。但遗憾的是 ,目前所有的研究均是在均匀风场或紊流风场中识别气动导数 ,而没有考虑暴雨的作用。本文采用随机系统识别方法 ,在模拟的风雨共同作用条件下识别了薄平板模型的气动导数。初步研究结果表明 ,风雨共同作用下平板结构的气动导数明显不同于仅有风作用的结果。这一新发现值得研究人员和工程技术人员的重视     

Estimation of torsional-flutter probability in flexible bridges considering randomness in flutter derivatives

Torsional-flutter instability is an aeroelastic phenomenon of interest to the bridge engineer, corresponding to a torsionally unstable vibration regime of the deck driven by wind excitation and appearing beyond a certain critical wind velocity. In this study a method for the derivation of the flutter probability for long-span bridges with bluff decks is proposed.In the first part of this study the deterministic problem is addressed. In contrast with the classical solution method in the frequency domain based on a numerical procedure for assessing the critical wind velocity, a single-mode “closed-form” algorithm for the derivation of the critical velocity was investigated. A polynomial representation of the aeroelastic-loading coefficients (flutter derivatives), necessary for torsional-flutter analysis, was utilized.In the second part an algorithm for estimating the torsional-flutter probability was developed, considering randomness in bridge properties, and flutter derivatives in particular due to their preeminent role in torsional-flutter velocity estimation.Experimental errors in the extraction of flutter derivatives from wind tunnel tests were analyzed. The “closed-form” algorithm, developed in the first part, allowed for a direct numerical solution of the flutter probability in a simple way.The torsional-flutter probability for three simulated bridge models with rectangular closed-box and truss-type girder deck was numerically determined. A set of experimental data, available from the literature, was employed. The simulations enabled the validation of the proposed algorithm.     

Relationships among aerodynamic admittance functions, flutter derivatives and static coefficients for long-span bridges

Wind actions on long-span bridges are commonly considered as the superimposition of buffeting forces and self-excited forces, depending on the aerodynamic admittance functions and on the flutter derivatives, respectively. Since bridge deck sections are bluff bodies, the aerodynamic admittance functions and the flutter derivatives have to be determined experimentally by wind tunnel tests. This paper introduces a generalized quasi-static theory, defining new relationships among the flutter derivatives and the aerodynamic admittance functions. All the relationships are theoretically verified for the zero circular frequency; based upon experimental results, the validation of the relationships among the flutter derivatives is also provided for non-zero values of the frequency.     

风雨联合作用下大跨桥梁颤振稳定性试验研究

针对风雨联合作用下的大跨桥梁颤振稳定性,以一开槽双箱梁桥梁为研究对象,通过在大气边界层风洞中搭建的风雨联合作用试验系统,完成基于自由振动法的节段模型颤振试验。通过分析不同雨强下该桥梁主梁的颤振导数以及颤振临界风速,进而获取降雨对大跨桥梁颤振稳定性的影响规律。试验结果显示:颤振导数随雨强变化而变化,其中体现扭转气动阻尼特性的颤振导数变化较为显著,随雨强增大,颤振临界风速先增大后减小。试验结果表明:降雨对大跨桥梁的颤振导数以及颤振临界风速均有一定影响。     

Time-domain and frequency-domain approaches to identification of bridge flutter derivatives

Flutter derivatives are essential for flutter analysis of long-span bridges, and they are generally identified from the vibration testing data of a sectional model suspended in a wind tunnel. Making use of the forced vibration testing data of three sectional models, namely, a thin-plate model, a nearly streamlined model, and a bluff-body model, a comparative study was made to identify the flutter derivatives of each model by using a time-domain method and a frequency-domain method. It was shown that all the flutter derivatives of the thin-plate model identified with the frequency-domain method and time-domain method, respectively, agree very well. Moreover, some of the flutter derivatives of each of the other two models identified with the two methods deviate to some extent. More precisely, the frequency-domain method usually results in smooth curves of the flutter derivatives. The formulation of time-domain method makes the identification results of flutter derivatives relatively sensitive to the signal phase lag between vibration state vector and aerodynamic forces and also prone to be disturbed by noise and nonlinearity.     

Direct identification of flutter derivatives and aerodynamic admittances of bridge decks

Flutter derivatives and aerodynamic admittances provide basis of predicting the critical wind speed in flutter and buffeting analysis of long-span cable-supported bridges. In this paper, one popular stochastic system identification technique, covariance-driven stochastic subspace identification (SSI in short), is first presented for estimation of the flutter derivatives and aerodynamic admittances of bridge decks from their random responses in turbulent flow. Numerical simulations of an ideal thin plate are adopted to extract these aerodynamic parameters to evaluate the applicability of the present method. Then wind tunnel tests of a streamlined thin plate model and a Π type blunt bridge section model were conducted in turbulent flow and the flutter derivatives and aerodynamic admittances are determined by the SSI technique. The identified aerodynamic parameters are compared with the theoretical ones and the results indicate the applicability of the current method.     

Flutter mechanism for rectangular prisms in smooth and turbulent flow

The aim of the present work is to clarify the flutter mechanism for suspended long span bridges via a parametric analysis on flutter instability for a set of given deck profiles. Several wind tunnel tests in the DIC-CRIACIV boundary layer wind tunnel (BLWT) have been carried out on spring suspended section models such as rectangular cylinders of different slenderness ratios B/D=5 and 12.5, where B is the longitudinal length of the prism and D is the height of the prism. The main experimental parameters needed for examining whether a given bridge profile is flutter-prone below a certain mean wind velocity are the flutter derivatives, so a system identification procedure (combined system identification method, CSIM) has been developed to extract simultaneously all flutter derivatives from two degrees of freedom (2DoF) section model test results (coupled vertical-torsional free vibration tests). The parametric analysis includes the investigation on (1) the effects of model dynamic properties on BLWT test results, (2) the consequence of turbulence on bridge stability, (3) the possible definition of an aerodynamic stability performance index (β) for rectangular cylinders for designing purposes.     

大跨度桥梁多模态耦合颤振二分法全自动搜索

在分析和评价国内外对大跨度桥梁风致颤振研究的成果和目前各种常用颤振分析方法优缺点的基础上,基于结构的固有模态坐标,进一步完善了大跨度桥梁多模态耦合颤振的双参数搜索状态空间法。通过将结构和气流作为一个动态系统,最后将颤振问题归结为数学上一个非对称实矩阵的广义特性值问题。在求解中,风速搜索采用了二分法加速收敛技术,频率采用搜索迭代法,该方法可以按指定的搜索精度,迅速搜索到颤振临界风速和颤振频率,无须设定步长,且能真实地给出各阶模态频率和阻尼比随风速的全过程。最后对具有理想平板截面的悬臂梁进行颤振分析验证了本文方法的正确性,同时对总跨度1177m的香港汀九大桥进行颤振分析,计算得到的颤振临界风速基本与风洞试验结果基本一致,进一步表明本文方法的可靠性和高效性。     

风驱雨作用下桥梁主梁颤振节段模型试验研究

针对风驱雨作用下桥梁主梁的颤振问题,依据风驱雨作用和主梁振动特点,给出了分别考虑雨滴冲击和表面积水后的降雨相似关系,并探讨了其选取原则。选取大跨度桥梁较常采用的典型断面,通过节段模型试验模拟了风驱雨对主梁断面的颤振导数和颤振发生过程的影响。试验结果表明:主梁断面的颤振气动导数随雨强的变化无明显规律,各导数的变化量值相当,随风速增加,降雨引起的导数变化有所加大,但基本没有改变其随风速变化的整体趋势,试验雨强120mm/h时,模型颤振临界风速会有20%～30%左右的提高,但考虑雨强相似比后可以认为降雨对桥梁主梁的风致颤振失稳特征的影响基本可以忽略。     

Data-driven stochastic subspace identification of flutter derivatives of bridge decks

Most of the previous studies on flutter derivatives have used deterministic system identification techniques, in which the buffeting forces and the associated responses are considered as noises. In this paper, one of the most advanced stochastic system identification, the data-driven stochastic subspace identification technique (SSI-DATA) was proposed to extract the flutter derivatives of bridge decks from the buffeting test results. An advantage of the stochastic method is that it considers the buffeting forces and the responses as inputs rather than as noises. Numerical simulations and wind tunnel tests of a streamlined thin plate model conducted under a smooth flow by the free decay and the buffeting tests were used to validate the applicability of the SSI-DATA method. The results were compared with those from the widely used covariance-driven SSI method. Wind tunnel tests of a two-edge girder blunt type of Industrial-Ring-Road Bridge deck (IRR) were then conducted under both smooth and turbulent flows. The identified flutter derivatives of the thin plate model based on the SSI-DATA technique agree well with those obtained theoretically. The results from the thin plate and the IRR Bridge deck helped validate the reliability and applicability of the SSI-DATA technique to various experimental methods and wind flow conditions. The results for the two-edge girder blunt type section show that applying the SSI-DATA yields better results than those of the SSI-COV. The results also indicate that turbulence tends to delay the onset of flutter compared with the smooth flow case.     

大跨度桥梁的抗风强健性及颤振评价

国际学术界公认的结构强健性,是指结构系统抵抗未曾遇见或超乎正常的环境作用的能力。为了突出风灾的未曾遇见性和超乎正常性,文章率先将结构强健性的理念引入桥梁抗风设计与颤振评价中,定义桥梁抗风强健性包括强度、刚度和稳定3个方面,并提出采用桥梁可以抵抗的设计风速的重现时间来表示抗风强健性。针对桥梁抗风稳定中的颤振强健性评价问题,建立用4个随机变量表示的颤振安全域度随机模型,提出采用等效中心点法来计算颤振强健性的可靠指标、失效概率和重现时间。基于提出的颤振强健性评价方法,对4座已经建成的大跨度桥梁和4座将要建设的大跨度桥梁进行颤振强健性分析。     

A methodology for the experimental extraction of indicial functions for streamlined and bluff deck sections

This paper addresses the experimental derivation of step-response non-stationary forces in the time domain (indicial functions), to be used in iterative numerical simulations (convolution techniques) for flutter instability analyses of suspension bridges, as an alternative to aeroelastic derivatives evaluated in the frequency domain. In particular a system, conceived for the direct extraction of these coefficients, has been designed, built and tested in wind tunnel. The study has investigated the experimental procedure of measurement and the statistical analysis of the recorded data. Both airfoil-type models (NACA) and a bluff section, simulating the behavior of bridge section, have been considered.     

Active aerodynamic stabilisation of long suspension bridges

The paper describes the addition of actively controlled aerodynamic appendages (flaps) attached along the length of the bridge deck to dampen wind-induced oscillations in long suspension bridges. A novel approach using control systems methods for the analysis of dynamic stability is presented. In order to make use of control analysis and design techniques, a linear model of the structural and aerodynamic motion around equilibrium is developed. The model is validated through comparison with finite element calculations and wind tunnel experimental data on the Great Belt East Bridge in Denmark. The developed active control scheme is local in that the flap control signal at a given longitudinal position along the bridge only depends on local motion measurements. The analysis makes use of the Nyquist stability criteria and an analysis of the sensitivity function for stability analysis. The analysis shows that the critical wind speed for flutter instability and divergence is increased substantially by active control.     

Identification of aerodynamic parameters for eccentric bridge section model

The equations of motion for bridge deck section model elastically suspended in wind tunnel are formulated about mass center of the system using the Lagrangian approach, accommodating both the elasticity and damping eccentricities in the formulation. The Subsection Extended-Order Iterative Least Square (SEO-ILS) algorithm is developed in the state space for direct identification of system matrices from free vibration data of section model obtained from wind tunnel testing. The flutter derivatives can be extracted straightforwardly from the difference in the system matrices identified at zero wind velocity and at a specific wind velocity, respectively. By making use of complex modal decomposition technique, a procedure is employed to correct the system matrix at zero wind velocity considering both eccentricities. The proposed method is applied to identify the flutter derivatives of a thin plate section model and the section model of a suspension bridge. The results show a favorable agreement between the flutter derivatives of a thin plate obtained with the proposed method and those derived from the analytic formulae. The identified direct flutter derivatives of the suspension bridge section model also are in good agreements with those obtained using Scanlan's method. It is shown that the use of the corrected system matrix at zero wind velocity leads to better accuracy in identifying the flutter derivatives especially at high reduced wind velocity than using the original system matrix, and the eccentricity is found to have more influence on the cross flutter derivatives than on the direct flutter derivatives.