Analysis of Long Span Bridge Response to Winds: Building Nexus between Flutter and Buffeting

Transcription

1 Analysis of Long Span Bridge Response to Winds: Building Nexus between Flutter and Buffeting Xinzhong Chen 1 Abstract: In current bridge aeroelastic analysis frameworks, predictions of flutter instability and buffeting response to wind fluctuations are treated as two separate procedures. The flutter instability is analyzed through the solution of a complex eigenvalue problem, offering information on how the self-excited forces influence bridge dynamics, especially the modal damping, and eventually drive the bridge to develop unstable flutter motion. On the other hand, the buffeting response is quantified through spectral analysis that involves the evaluation of a complex frequency response matrix/transfer matrix at discrete frequencies, offering prediction of the statistics of bridge response. The modal properties at varying wind velocities predicted through flutter analysis have not yet been explicitly employed in predicting and interpreting buffeting response. Furthermore, current buffeting analysis frameworks often regard the complex aerodynamic admittance functions as real-valued, neglecting the phase differences of buffeting force components with respect to wind fluctuations. In this paper, a unified analysis framework of integrating both flutter and buffeting analysis is presented. The buffeting response is explicitly expressed in terms of the bridge modal properties that are influenced by the self-excited forces and predicted through the flutter analysis. Closed-form formulations are provided for quantifying the variance/covariance of buffeting response. This framework not only is computationally more effective, but also sheds more physical insight into bridge aeroelastic response by explicitly linking the effects of self-excited and buffeting forces on bridge response. The effectiveness and accuracy of the proposed framework are illustrated through a long span suspension bridge example. The multimode coupled bridge aeroelastic response is discussed with an emphasis placed on the significance of complex aerodynamic admittance functions on buffeting response. DOI: / ASCE : CE Database subject headings: Flutter; Buffeting; Aerodynamics; Aeroelasticity; Bridges; Wind loads; Vibration. Introduction Bridge response to strong winds is one of the critical issues to be addressed in long span bridge design and construction. Wind forces on bridge sections are modeled in terms of drag, lift, and pitching moment in the alongwind, acrosswind, and torsional directions. These forces are generally separated into static mean, self-excited motion-induced, and buffeting turbulence-induced force components, which are featured respectively by static force coefficients, flutter derivatives, and aerodynamic admittance functions e.g., Davenport 1962; Scanlan 1978a,b, 1993; Chen and Kareem 2002a. These force functions can be experimentally quantified through wind tunnel tests using bridge section models. Furthermore, to quantify integrated wind forces on a finite element and the entire structure, the spanwise correlation of forces need to be considered. While the self-excited forces are usually regarded as spatially fully correlated, the buffeting forces are modeled as partially correlated. The lack of spatial correlation of buffeting forces leads to a reduction in the integrated force as 1 Assistant Professor, Wind Science and Engineering Research Center, Dept. of Civil and Environmental Engineering, Texas Tech Univ., TX xinzhong.chen@ttu.edu Note. Associate Editor: Sashi K. Kunnath. Discussion open until May 1, Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on June 9, 2005; approved on April 19, This paper is part of the Journal of Structural Engineering, Vol. 132, No. 12, December 1, ASCE, ISSN /2006/ /$ compared to the fully correlated case. This reduction effect is described by the joint acceptance function e.g., Davenport 1962, 1999, Chen and Kareem 2002a. Wind-induced bridge response can be evaluated utilizing analysis frameworks that integrate information regarding wind climate at the bridge site, aerodynamic forces on the bridge sections, and finite-element modeling of the entire bridge. Under wind excitation, a bridge experiences three-dimensional static deformation due to the action of static wind forces. Evaluation of static deformation often requires the use of a nonlinear iterative analysis framework, taking into account both structural nonlinearity associated with large deflection and nonlinear dependence of wind forces on structural deflection, especially on the rotation of a bridge deck contributing to a change in the angle of incidence and thereby in force characteristics. On the other hand, the analysis of dynamic response around the static equilibrium can be carried out by using a linear framework based on the equations of motion in terms of structural modal coordinates Scanlan 1978a,b. The presence of self-excited forces leads to additional aerodynamic stiffness and damping terms for the bridge system, which are frequency dependent due to the unsteady nature of wind-bridge interaction. The contribution of aerodynamic stiffness and damping terms renders the equations of bridge motion to be frequency dependent and coupled. The intermodal aerodynamic coupling becomes significant for longer spans at higher wind velocities. This necessitates the use of multimode coupled analysis frameworks for response prediction rather than the traditional mode-by-mode approach, which neglects the intermodal coupling e.g., Irwin 1998; Jones et al. 1998; Yamada et al. 1999; Chen et al. 2000a,b; 2004; Ge and Tanaka 2000; Xu et al / JOURNAL OF STRUCTURAL ENGINEERING ASCE / DECEMBER 2006

2 Within these multimode coupled analysis frameworks, the flutter analysis is performed through solution of a complex eigenvalue problem. This analysis not only offers the prediction of critical flutter velocity, beyond which one of bridge modes becomes negatively damped, but also provides information regarding the influence of self-excited forces on the modal properties as wind velocity increases. This helps in understanding how the selfexcited forces change bridge dynamics, particularly the modal damping, and eventually drive the bridge to flutter instability. On the other hand, the buffeting analysis is often conducted in the frequency domain through spectral analysis, while the time domain frameworks are more appropriate for nonlinear problems introduced by either structural or aerodynamic origins, and for bridges under nonstationary wind excitations Diana et al. 1999; Chen et al. 2000b, Chen and Kareem 2003a. The frequency domain buffeting analysis involves evaluation of the complex frequency response matrix/transfer matrix of the bridge aeroelastic system. For a linear system with frequency independent parameters, the transfer matrix is readily expressed in terms of the modal properties of the system. However, this is not the case for a bridge aeroelastic system involving frequency dependent parameters. Consequently, the information on modal properties at varying wind velocities predicted through flutter analysis has not yet been explicitly employed in predicting and interpreting buffeting response. As a result, in current analysis frameworks, the predictions of flutter and buffeting have to be treated as two separate procedures without establishing an explicit nexus between them. This practice has limited our ability to better understand the aeroelastic behavior of long span bridges to strong winds. The flutter derivatives describe the frequency dependent unsteady characteristics of the self-excited forces concerning the amplitudes and phase differences of forces with respect to bridge motion. These unsteady descriptions have been completely modeled in flutter analysis. Similar information regarding buffeting forces is provided by the complex aerodynamic admittance functions. However, current buffeting analysis frameworks often consider these functions as real-valued, neglecting the phase differences of buffeting forces with respect to wind fluctuations. This approximation may be acceptable for bridges with weak aerodynamic intermodal coupling, but may have notable influence on the accurate response prediction of bridges with strong intermodal coupling. A preliminary discussion of the role of complex aerodynamic admittance functions on buffeting response of a bridge deck has been made by Diana et al In this paper, a unified framework of integrating the multimode coupled flutter and buffeting analysis is presented. The complex frequency response matrix of the bridge aeroelastic system is explicitly expressed in terms of its complex modal properties, which are evaluated through flutter analysis. Closed-form formulations for quantifying the variance/covariance of buffeting response are presented. The effectiveness and accuracy of this framework are illustrated utilizing a long span suspension bridge example. Within this framework, the multimode coupled flutter and buffeting responses are discussed, focusing on the significance of complex aerodynamic admittance functions on the buffeting response. Equations of Motion A long span bridge subjected to wind excitations with the mean wind direction normal to the bridge deck axis is considered. The dynamic displacements of the bridge deck at the spanwise location x in the acrosswind, alongwind, and torsional directions are denoted by h x,t downward, p x,t downwind, and x,t nose-up, respectively. The static, self-excited, and buffeting forces acting on the bridge deck per unit length, i.e., lift downward, drag downwind, and pitching moment nose-up, are given Scanlan 1978a,b, 1993; Jones et al. 1998; Chen et al. 2002a,b as L s x = 1 2 U2 BC L ; D s x = 1 2 U2 BC D ; M s x = 1 2 U2 B 2 C M L se x,t = 1 2 U2 * 2b kh ḣ 1 U + kh *b 2 U + k2 H * 3 + k 2 * H h 4 b + kh 5 * ṗ U 1 + k 2 H 6 * p b 2 D se x,t = 1 2 U2 * 2b kp ṗ 1 U + kp *b 2 U + k2 P * 3 + k 2 * P p 4 b + kp 5 * ḣ U + k 2 P 6 * h b 3 M se x,t = 1 2 U2 2b 2 * ka ḣ 1 U + ka * b 2 U + k2 A * 3 + k 2 * A h 4 b + ka * ṗ 5 U + k 2 A 6 * p b 4 L b x,t = 1 u 2 U2 B 2C L Lu U + C L + C w D Lw U D b x,t = 1 u 2 U2 B 2C D Du U + C D C w L Dw U M b x,t = 1 2 U2 B 2 2C u M Mu U + C M w Mw U where =air density; U=mean wind velocity; B=2b=bridge deck width; k= b/u=reduced frequency; =frequency of motion; C L, C D, C M =static force coefficients; C L =dc L /d ; C D =dc D /d ; C M =dc M /d ; H * j, P * * j, and A j,2,...,6 =flutter derivatives; Lu, Lw, Du, Dw, Mu, and Mw =aerodynamic admittance functions; u and w=wind fluctuations in the longitudinal and vertical directions; and subscripts s, se, and b designate the static, self-excited, and buffeting force components, respectively. It is noted that the flutter derivatives and aerodynamic admittance functions are functions of reduced frequency. The aerodynamic forces acting on other bridge elements, i.e., towers and cables, can also be provided using similar formulations. Following the finite-element procedure and the modal analysis approach, the equations of bridge motion can be expressed as follows in terms of the generalized modal displacements q e.g., Chen et al. 2000a,b : Mq + Cq + Kq = Q se + Q b M = diag m j ; C = diag 2 sj sj m j ; K = diag m j 2 sj JOURNAL OF STRUCTURAL ENGINEERING ASCE / DECEMBER 2006 / 2007

3 Q se = 1 2 U2 A s k q + b U A d k q Q b = 1 2 U2 A bu k u U + A bw k w U where M, C, and K=generalized mass, damping, and stiffness matrices; m j, sj, and sj = jth generalized mass, damping ratio, and frequency,2,...,m; M =total number of structural modes included ; Q se and Q b =generalized self-excited and buffeting force vectors; A s k and A d k =aerodynamic stiffness and damping matrices, which are functions of flutter derivatives and structural mode shapes; A bu k and A bw k =buffeting force matrices, which are functions of static force coefficients, aerodynamic admittance functions, coherence functions of aerodynamic forces, and structural mode shapes; and u and w = longitudinal and vertical component vectors of wind fluctuations. It is worthy of mention that, as the aerodynamic stiffness and damping matrices are frequency dependent and nondiagonal matrices, the equations of motion of the bridge aeroelastic system become frequency dependent and coupled. Multimode Coupled Flutter Analysis The equations of bridge motion can be represented in the statespace format Ẏ = A k Y + BQ b 12 Y = q q ; A k = 0 I M 1 K 1 k M 1 C 1 k ; B = 0 M 1 13 C 1 k = C 1 2 UbA d k ; K 1 k = K 1 2 U2 A s k 14 The influence of self-excited forces on the bridge modal properties can be quantified through the solution of the following complex eigenvalue problem: j j = A k j j 15 which is alternatively expressed as j 2 M + j C j k j + K 1 k j j = 0 j = j j + i j 1 j 2 ; j = j j j where j and j = jth complex eigenvalue and eigenmode mode shape with their complex conjugates j+m = j * and j+m = j * ; j and j = jth complex modal frequency and damping ratio; k j = j b/u; i= 1; and superscript * denotes the complex conjugate operator. It is noted that, as the system involves frequency dependent parameters, the solution of each eigenvalue requires an iterative calculation procedure until the assumed frequency used to evaluate the aerodynamic stiffness and damping matrices agrees with the predicted modal frequency. The predicted modal frequencies and damping ratios at varying wind velocities offer important information concerning the modifications in modal properties attributed to the self-excited forces. The predicted mode shapes provide information regarding the intermodal coupling in terms of the amplitude ratios and phase differences between different structural modal components. The existence of phase differences between different motion components distinguishes the complex modes from real-valued structure modes at zero wind velocity in which the structural motion components are always in phase or out of phase. The critical flutter velocity is determined when one of the modal damping ratios becomes zero. The corresponding complex mode shape characterizes the bridge motion in flutter. Multimode Coupled Buffeting Analysis Traditional Framework Based on the random vibration theory, the power spectral density PSD matrix of the generalized displacement q and the PSD function of any response component of interest, i.e., displacement, acceleration, and member force, v=d T q, are expressed as S q = H q * S Qb H q T S v = D T S q D H q = 2 M + i C 1 k + K 1 k 1 S Qb = U2 1 2 A * 2 bu k S u A T bu k /U A * bw k S w A T bw k /U 2 21 where S Qb, S u, and S w =power spectral density matrices of Q b, u, and w, respectively; D=modal participation coefficients to response v; H q =complex frequency response matrix/transfer matrix; and superscript T denotes the matrix transpose operator. While the cross-power spectra between longitudinal and vertical wind fluctuations are neglected in Eq. 21, those can be readily included in a straightforward manner. The mean square value of the response is then quantified by integrating its power spectrum over the frequency range of interest. The expected maximum response is determined by its root-mean-square RMS value multiplied by a peak factor, which usually ranges between 3 and 4. It is emphasized that, while this currently used buffeting analysis framework is straightforward, it offers less information on how the modal properties at varying wind velocities affect the buffeting response. In the following, a unified framework for buffeting analysis is presented in which the transfer matrix is directly calculated based on the modal properties obtained from the flutter analysis. This framework establishes a clear nexus between multimode coupled bridge flutter and buffeting. New Framework The frequency response matrix and the response spectra generally have peaks around the modal frequencies. Subsequently, the integration of response spectra in close neighborhoods of modal frequencies dominates the response variance and covariance. Within the vicinity of each modal frequency, the frequency dependent aerodynamic stiffness and damping terms can be approximately regarded as constants by taking their values at the modal frequency. This approximation is generally acceptable because it is applied only in a very small range of frequencies, and also the aerodynamic parameters rarely markedly vary with frequency. For example, around the nth complex modal frequency, n, the bridge aeroelastic system can be approximately regarded as a fre / JOURNAL OF STRUCTURAL ENGINEERING ASCE / DECEMBER 2006

4 quency independent linear system with K 1 k K 1 k n and C 1 k C 1 k n, where k n = n b/u. Consequently, the equations of motion in the frequency domain around the nth modal frequency can be expressed in the following state-space format: i Y = A k n Y + BQ b 22 where Y =Fourier transform of Y t. n The jth complex eigenvalue and mode shape of A k n, i.e., j n and j where the superscript n denotes the values corresponding to the system matrix defined at k n, are given by n j n n j = A k n j 23 n j = n j n n j + i 1 j n j 2 ; n j = n j n j n 24 j The corresponding eigenmode of A T k n, also referred to as the left eigenmode of A k n, is expressed as n j n j = A T n k n j 25 n j = ju n 26 jd The eigenmodes of A k n and A T k n satisfy the orthogonality condition n j T n l = jl which is alternatively expressed as 27 n jd T M 1 2 j + M + C 1 k j n l = jl 28 where jl =the Kronecker delta, i.e., when l= j, lj =1; otherwise, lj =0. By decomposing the response into complex modal response components, the transfer matrix of the linear system described by Eq. 22 can be expressed in terms of its modal properties Soong and Grigoriu 1993; Chen and Kareem 2002b as M H q = n j n j T /m j n + n j * n j *T /m j i j i n j * 29 where n j = m j M 1 n jd 30 It is important to note that each of the complex eigenvalues and mode shapes obtained from the flutter analysis, i.e., j and j,2,...,m, are determined using the system matrix defined at the individual modal frequency, i.e., A k j, which indicates that all eigenvalues and mode shapes are not determined using an identical system matrix. Therefore, n j = j, n j = j, and n j = j for j=n; n j j, n j j, and n j j for j n j,n=1,2,...,m, where j is determined from the eigenvalue problem of A T k j associated with j T j 2 j M + j C 1 k j + K 1 k j = 0 and is normalized as 31 T j 2 j M + C 1 k j j = m j 32 Eq. 29 clearly points at the contribution of each modal component to the transfer matrix. It is noted that, at the vicinity of each modal frequency, the transfer matrix is dominated by the contributions of the modal components with frequencies close to this modal frequency. The contributions of other modal components are of less significance. By respectively replacing n j, n j, n and j with j, j, and j,2,...,m, Eq. 29 can be further simplified as M H q = j T j /m j + * j *T M j /m j * i j i j = H j0 i E j + F j 33 E j = j j T + j * j *T ; F j = j j T j * + j * j *T j 34 1 H j0 = m j 2 j i2 j j These formulations lead to the evaluation of the transfer matrix directly based on the complex modal properties that are obtained from the flutter analysis. These formulations are expected to offer good approximation of the transfer matrix in the frequency range around all modal frequencies, which is of importance for the evaluation of response. The new format not only is computationally more effective than the traditional framework, which requires calculation of the complex inverse matrix at each discrete frequency Eq. 20, but also sheds more physical insight to bridge aeroelastic response by explicitly linking the effects of the self-excited and buffeting forces. The self-excited forces affect the modal properties of the system, thus influencing the way the bridge responds to wind fluctuations. Obviously, when wind velocity exceeds the critical flutter velocity, the contribution of the flutter mode becomes infinite, and thus the system becomes unstable. It has been noted that a similar approximation regarding the dependence of complex modal properties on frequency has been made in the techniques for identification of flutter derivatives e.g., Chen and Kareem This approximation allows the extraction of flutter derivatives from the free vibration records of a spring-mounted two-degree-of-freedom bridge section model. It is also worth mentioning that the framework proposed herein is more straightforward as compared to the procedure presented by Chen and Kareem 2002b, in which j,2,...,m was given by the jth column of =m j M 1 V T R, where V R is the right half of 1 = M 1. Using the transfer matrix in terms of the complex modal properties, the PSD matrix of the modal response becomes M M S q = H * j0 H l0 S Qbc jl l=1 S Qbc jl = i E j + F j S Qb i E l T + F l T When the aerodynamic intermodal coupling is negligible, the complex modes reduce to the real-valued structural modes, which leads to the formulations for the uncoupled system: j = lj = lj ; E j = 0; H q = diag H j0 ; j = lj = i 2 j 1 j 2 lj 38 F j = F j rs = rsj S Qbc jl = S Qjl jl JOURNAL OF STRUCTURAL ENGINEERING ASCE / DECEMBER 2006 / 2009

5 S q = S qjl = H * j0 H l0 S Qjl 41 Y j = H j0 Q b 49 where S Qjl =cross-power spectral density function between the jth and lth generalized forces, i.e., Q j t and Q l t ; and rsj =the Kronecker delta, i.e., when r=s= j, rsj =1; otherwise, rsj =0. Closed-Form Formulation By assuming that the transfer matrix is narrow-banded and the force spectra are constants around the modal frequencies, the covariance matrix, R q, can be quantified in a closed form Der Kiureghian 1980; Chen and Kareem 2005 R q = 0 I jj = j ; 4 j jl = M M Re S q d = I jj I ll jl Re S Qbc jl jl / K j K l l=1 42 I ll = l ; K j = m j 2 4 j ; K l = m l 2 l 43 l 3/2 8 j l jl j + l jl 1 2 jl 2 +4 j l jl 1+ 2 jl +4 2 j + 2 l jl S Qbc jl jl = 0.5 S Qbc jl j + S Qbc jl l where jl = j / l with 0 jl 1; jj = ll =1; and jl = lj 1 when j and l are well separated; and Re=real operator. This format of modal response combination corresponds to the complete quadratic combination CQC approach. It is noted that, in the derivation of the preceding closed-form formulations, the power spectra are regarded as one-sided spectra with respect to the circular frequency. When the power spectra are defined with respect to the natural frequency f = / 2, j and l in I jj and I ll should be replaced by f j and f l, respectively. It is emphasized that the correlation coefficient of the two modal response components depends on the frequency ratio, damping ratios, and intermodal coupling as well. In cases where the modal frequencies are well separated, i.e., jl = lj 0 j l, then M R q = I 2 2 jj Re S Qbc jj j /K j 46 This format of modal response combination corresponds to the square root of the sum of squares SRSS combination approach. The modal responses remain correlated due to the aerodynamic coupling. In cases where the aerodynamic intermodal coupling is negligible, then R q = R qjl = I jj I ll jl Re S Qjl jl / K j K l 47 Time Domain Simulation Based on the transfer matrix in terms of the modal properties, the modal response can be expressed in the frequency domain as M q = i E j Y j + F j Y j 48 and consequently in the time domain as M q t = E j Ẏ j t + F j Y j t 50 where Y j t is given as follows for its element Y n j t n=1,2,...,m : Ÿ j n t +2 j j Ẏ j n t + 2 j Y j n t = Q n t /m j 51 These formulations can be used for time domain simulation of bridge response to wind fluctuations without generating the time histories of self-excited forces. The influence of self-excited forces on buffeting response is accounted for through the complex modal properties. The time histories of buffeting forces can be either generated based on their spectra or calculated based on the time histories of wind fluctuations. In the latter format, the rational function approximation technique can be utilized to model the frequency dependent features of the unsteady buffeting forces instead of using the quasi-steady assumption Chen and Kareem 2002a, b. The time domain simulation technique with the generation of time histories of the self-excited and buffeting forces is more appropriate when aerodynamic nonlinearities are taken into consideration Chen and Kareem 2003a. Role of Complex Aerodynamic Admittance Functions As shown in Eq. 47, in cases where the aerodynamic intermodal coupling is negligible, the cross-power spectral density XPSD function of the generalized forces only influences the covariance/ correlation of modal responses. For bridges with well separated modal frequencies, the correlation of the modal responses becomes negligible. Subsequently, only the PSDs of the generalized forces are required for estimating the bridge response contributed by each mode independently. However, as shown in Eqs. 42 and 46, in cases where the aerodynamic intermodal coupling is noticeable, not only the PSDs but also the XPSDs of the generalized forces affect the variance and covariance of modal responses, even for bridges with separated modal frequencies. As the intermodal coupling is complexvalued, the XPSDs of the generalized forces must be modeled as complex-valued for an accurate modeling of their influences. To highlight this important issue, a bimodal coupled bridge aeroelastic system that involves the fundamental vertical and torsional modes under the excitation of only the w-component of wind fluctuation is considered as follows. The fundamental vertical mode shape is denoted as h 1 x 0, p 1 x =0, 1 x =0 in terms of the bridge displacements in the vertical, lateral, and torsional directions. The fundamental torsional mode shape is given by h 2 x =0, p 2 x 0, 2 x 0. Accordingly, the bridge displacements in the vertical and torsional directions are h x,t =h 1 x q 1 t and x,t = 2 x q 2 t, where q 1 t and q 2 t are the generalized coordinates. For simplicity and without loss of generality, only the lift and pitching moment acting on the bridge deck sections are considered. Therefore, the PSD matrix of the generalized forces is given by S Q1 = 1 2 B 2 U2 C L + C D 2 G h1 h 1 Lw k 2 J Lw k 2 S w /U / JOURNAL OF STRUCTURAL ENGINEERING ASCE / DECEMBER 2006

6 Fig. 1. Structural mode shapes in terms of bridge deck displacements: a Mode 1; b Mode 2; c Mode 8; d Mode 9; and e Mode 10 S Q2 = U2 B C M 2 G 2 2 Mw k 2 J Mw k 2 S w /U 2 53 * and their cross-spectrum, i.e., S Q12 =S Q21, is expressed as S Q12 = U2 1 2 B 3 C 2 L + C D C M G h1 2 * Lw k Mw k J LMw k 2 S w /U 2 54 where J Lw k 2, J Mw k 2, and J LMw k 2 =joint acceptance functions given as follows by assuming that the coherence functions of the buffeting forces are identical to that of the wind fluctuations, i.e., coh w x 1,x 2, : J Lw k 2 h 1 x 1 h 1 x 2 coh w x 1,x 2, dx 1 dx 2 /G h1 h 1 = span span J Mw k 2 2 x 1 2 x 2 coh w x 1,x 2, dx 1 dx 2 /G 2 2 = span span Fig. 2. Modal frequencies versus wind velocity Fig. 3. Modal damping ratios versus wind velocity JOURNAL OF STRUCTURAL ENGINEERING ASCE / DECEMBER 2006 / 2011

7 J LMw k 2 h 1 x 1 2 x 2 coh w x 1,x 2, dx 1 dx 2 /G h1 2 = span span coh w x 1,x 2, = S w x 1,x 2, / Sw x 1, S w x 2, By further assuming J Lw k 2 = J Mw k 2 = J LMw k 2 = J w k 2, Lw k = w k, and Mw k = w k exp i w where w =phase angle between pitching moment and lift associated with the w-component of the wind fluctuations, the PSD matrix of the generalized forces then becomes S Q = B 2 1 C L + C D 2 2 G 2 U2 w k 2 J w k 2 h1 h 1 B C L + C D C M G h1 2 exp i w S w /U B C L + C D C M G h1 2 exp i w B 2 C M 2 G S Qbc By denoting i E j +F j =G, Eq. 45 then gives jj =GS Q G *. For instance, its element becomes jj S Qbc11 = G 11 2 S Q11 +2Re G 11 G * 12 S Q12 + G 12 2 S Q22 60 Similar formulations associated with the u-component of the wind fluctuations can be provided. These expressions clearly demonstrate that the phase differences between buffeting force components, introduced by the complex aerodynamic admittance functions, will influence the cross spectra of the generalized forces, and thereby potentially influence the multimode coupled buffeting response. Demonstration and Discussion Suspension Bridge Example A suspension bridge with a center span of nearly 2,000 m was used as an example to demonstrate the accuracy of the proposed framework and to discuss the multimode coupled bridge aeroelastic response. For simplicity and without loss of generality, only the aerodynamic forces acting on the bridge deck sections were considered. The von Karman spectra were used with turbulence intensities of 10 and 5% and integral length scales of 80 and 40 m, for the u- and w-components, respectively. The aerodynamic admittance functions used were based on Davenport s formula for drag with a decay factor of 8 and the Sears function for lift and pitching moment. These aerodynamic admittance functions are modeled as real-valued functions, thus implicitly assuming that the buffeting forces are in phase with the wind fluctuations. The influence of complex aerodynamic admittance functions will be addressed in a later part of this study. Two different coherence functions were used for the buffeting force components associated with the u- and w-components, respectively. The drag component of the self-excited forces was evaluated based on the quasi-steady assumption. The lift and pitching moment components were calculated based on the flutter derivatives derived from the Theodorsen function. The static force coefficients were C D =0.3230, C L =0.0942, C M =0.0104, dc D /d =0, dc L /d =1.905, and dc M /d = The first 15 natural modes with frequencies ranging from 0.03 to 0.2 Hz were considered to describe the bridge dynamics. As shown in Fig. 1, modes 1, 2, and 10 are the fundamental symmetric modes in the lateral, vertical, and torsional directions, respectively. Modes 8 and 9 are the second symmetric vertical and lateral modes. The lateral and torsional modes have both lateral and torsional motions due to structural coupling, while the vertical modes consist of almost pure vertical motion. Complex Modal Properties and Coupled Flutter The modal frequencies and damping ratios at varying wind velocities are shown in Figs. 2 and 3. At zero wind velocity, those Fig. 4. Complex mode shapes in terms of amplitudes and phase angles of structural modal coordinates U=60 m/s : a complex Mode Branch 10, 10 ; b left complex Mode Branch 10, 10d 2012 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / DECEMBER 2006

8 Fig. 5. Flutter mode shape in terms of amplitudes and phase angles of structural modal coordinates U=69.3 m/s are the structural modal frequencies and damping ratios. At low wind velocities in which the intermodal coupling is negligible, the complex mode shapes are almost identical to those of the corresponding real-valued structural modes. The influence of the selfexcited forces is primarily to modify the modal damping ratios. Conversely, at high wind velocities, the intermodal coupling becomes significant. Fig. 4 a shows the complex modal shapes of modal branch 10 at the wind velocity of 60 m/s, in terms of amplitudes and phase angles of the structural modal coordinates. The corresponding mode shape of the system matrix transpose is shown in Fig. 4 b. Mode branch 10 contains significant coupled motions of structural modes 2, 8, 9, and 10 with phase differences. As shown in Figs. 2 and 3, around the wind velocity of 65 m/s, the modal frequencies of modal branches 9 and 10 are close to each other, which leads to curve veering of the frequency and damping loci Chen and Kareem 2003b. As a result, these two complex modes exchange their properties after the curve veering. The damping of modal branch 9 becomes negative when wind velocity exceeds the critical flutter velocity of 69.3 m/ s. As shown in Fig. 5, the flutter motion is composed of multimodes, with dominant participation of structural modes 2, 8, 9, and 10. The coupling between modes 2 and 10 is most significant, while notable participation of structural modes 8 and 9 is observed, which is attributed to their frequencies being close to the flutter modal frequency. Fig. 6. Complex frequency response functions relevant to structural modes 2 and 10: a Mode 2; b Mode 10 Fig. 7. Displacement power spectra of bridge deck at midpoint of center span U=20 m/s : a lateral displacement; b vertical displacement; and c torsional displacement JOURNAL OF STRUCTURAL ENGINEERING ASCE / DECEMBER 2006 / 2013

9 Fig. 8. Displacement power spectra of bridge deck at midpoint of center span U=60 m/s : a lateral displacement; b vertical displacement; and c torsional displacement Comparison of Buffeting Response The buffeting analysis is conducted by using two schemes with different quantification of the frequency response matrix. The scheme denoted by Exact indicates the traditional approach, in which the frequency response matrix is directly calculated as the inverse of the frequency dependent system matrix at each discrete frequency Eq. 20, whereas Approx. represents that based on the modal properties as suggested in this study. The comparison of the transfer functions relevant to structural modes 2 and 10 at the wind velocities of 20, 40, and 60 m/s is shown in Fig. 6. The response spectra of the bridge deck displacements at the midpoint of the center span at 20 and 60 m/s are portrayed in Figs. 7 and 8. The RMS values of the bridge deck displacements at different spanwise locations are shown in Fig. 9. The comparison of the closed-form solution with that by the traditional approach is shown in Fig. 10. These results demonstrate that the proposed scheme provides sufficiently accurate prediction of the buffeting Fig. 9. RMS displacements of bridge deck at different spanwise locations: a lateral displacements; b vertical displacements; and c torsional displacement response as compared to the traditional approach. The proposed scheme is computationally more effective and offers an clear nexus between the buffeting response and modal properties evaluated through flutter analysis. This helps in developing a better understanding of multimode coupled bridge response to strong winds. As shown in Fig. 11 for the response spectra at the wind velocity of 69 m/s, with an increase in wind velocity, the damping ratio of the modal branch with a frequency of around 0.12 Hz decreases and its contribution to bridge response becomes even more dominant. This leads to flutter instability beyond 69.3 m/s due to negative damping. Significance of Different Wind Fluctuations In order to investigate the significance of different components of wind fluctuations on the multimode coupled buffeting response, the buffeting response components associated with the u- and 2014 / JOURNAL OF STRUCTURAL ENGINEERING ASCE / DECEMBER 2006

10 Fig. 10. Comparison of RMS displacements of bridge deck at different spanwise locations: a lateral displacement; b vertical displacement; and c torsional displacement w-components of wind fluctuations are quantified separately. The RMS values of total response due to the combined action of both the u- and w-components are given as the square root of the sum of squares of those u- and w-components. Figs. 12 and 13 show the results of the RMS values of the bridge deck displacements at 20 and 60 m/s, respectively. Based on the values of the static force coefficients and the formulations of the buffeting forces, it is expected that the drag force on the bridge deck sections is primarily associated with the u-component of the wind fluctuations, while both lift and pitching moment are associated with the w-component. As the lateral motion is involved in both the lateral and torsional modes, the action of drag force associated with the u-component of the wind fluctuations dominates both the lateral and the torsional response components. It is interesting to note that the vertical displacement is dominated by the action of the lift force associated with the w-component at 20 m/s, but with markable contribution of the u-component at a wind velocity of 60 m/s. Thus again attests to the aerodynamic coupling effect between the vertical and torsional motions. It also should be Fig. 11. Displacement power spectra of bridge deck at midpoint of center span U=69 m/s : a lateral displacement; b vertical displacement; and c torsional displacement mentioned that the relative significance of the wind fluctuations may be influenced by the spectral characteristics of the wind fluctuations. Influence of Complex Aerodynamic Admittance Functions In the preceding calculations, the aerodynamic admittance functions were modeled as real-valued functions, as is generally assumed in the literature. In order to highlight the influence of the complex aerodynamic admittance functions on the buffeting response, the XPSD of the generalized forces of modes 2 and 10 is given as complex-valued, i.e., S Q2,10 = S Q2,10 exp i w, * S Q10,2 =S Q2,10, where the phase angle w = 20, 0, and 20, and w =0 corresponds to the aforementioned traditional analysis. Fig. 14 shows its influence on the predicted torsional dis- JOURNAL OF STRUCTURAL ENGINEERING ASCE / DECEMBER 2006 / 2015

11 Fig. 12. Significance of wind fluctuations on buffeting response U=20 m/s : a lateral displacement; b vertical displacement; and c torsional displacement placement of the bridge deck associated with the w-component of the wind fluctuations at a wind velocity of 20 and 60 m/s. Results show that the phase differences of the buffeting lift and pitching moment may exhibit markable influence on the bridge response when intermodal coupling is significant, while this has less influence on bridge response when the intermodal coupling is negligible. Concluding Remarks Fig. 13. Significance of wind fluctuations on buffeting response U=60 m/s : a lateral displacement; b vertical displacement; and c torsional displacement The unsteady nature of the bridge-wind interaction renders the equations of motions of a bridge aeroelastic system in terms of structural modal responses to be frequency dependent and coupled. Current analysis frameworks for bridge aeroelastic analysis have to deal with the predictions of flutter and buffeting analysis as two separate procedures, without establishing an explicit nexus. The bridge modal properties at varying wind velocities, which are influenced by the self-excited forces and predicted through flutter analysis, have not yet been employed in the quantification and interpretation of the buffeting response. The buffeting analysis conducted in the frequency domain has to rely on the evaluation of a complex frequency response matrix at discrete frequencies as the inverse of the system matrix, involving frequency dependent parameters. This practice not only is computationally ineffective, but also has limited our understanding of how a bridge responds to strong wind excitations. This study presented a unified framework that integrated the analysis of multimode coupled flutter and buffeting. The buffeting response was explicitly expressed in terms of bridge modal properties at varying wind velocities with closed-form solutions for the variance and covariance of the bridge response. The effectiveness and accuracy of the proposed framework was illustrated by the response analysis of a long span suspension bridge example. The results revealed that the proposed framework clearly high / JOURNAL OF STRUCTURAL ENGINEERING ASCE / DECEMBER 2006

12 Fig. 14. Influence of complex aerodynamic admittance functions on RMS torsional displacements of bridge deck: a U=20 m/s; b U=60 m/s lighted the role of self-excited and buffeting forces on bridge aeroelastic response. Using the proposed framework, the significance of different wind fluctuations on the bridge buffeting response was addressed, which helped better our understanding of the role of intermodal coupling on the buffeting response. This study also pointed out the potentially important role of consideration of complex aerodynamic admittance functions for an accurate prediction of the coupled buffeting response. Acknowledgments The support of the new faculty startup funds provided by the Texas Tech University is gratefully acknowledged. References Chen, S. R., Cai, C. S., Chang, C. C., and Gu, M Modal coupling assessment and approximated prediction of coupled multimode wind vibration of long-span bridges. J. Wind. Eng. Ind. Aerodyn., 92, Chen, X., and Kareem, A. 2002a. Advances in modeling of aerodynamic forces on bridge decks. J. Eng. Mech., , Chen, X., and Kareem, A. 2002b. Advanced analysis of coupled buffeting response of bridges: A complex modal decomposition approach. Probab. Eng. Mech., 17 2, Chen, X., and Kareem, A. 2003a. Aeroelastic analysis of bridges: Turbulence effects and aerodynamic nonlinearity. J. Eng. Mech., 129 8, Chen, X., and Kareem, A. 2003b. Curve veering of eigenvalue loci of bridges with aeroelastic effects. J. Eng. Mech., 129 2, Chen, X., and Kareem, A Efficacy of the implied approximation in the identification of flutter derivatives. J. Struct. Eng., , Chen, X., and Kareem, A Coupled dynamic analysis and equivalent static wind loads on buildings with three-dimensional modes. J. Struct. Eng., 131 7, Chen, X., Matsumoto, M., and Kareem, A. 2000a. Aerodynamic coupling effects on flutter and buffeting of bridges. J. Eng. Mech., 126 1, Chen, X., Matsumoto, M., and Kareem, A. 2000b. Time domain flutter and buffeting response analysis of bridges. J. Eng. Mech., 126 1, Davenport, A. G Buffeting of a suspension bridge by storm winds. ASCE J. Struct. Div., 88 3, Davenport, A. G The missing links. Proc., 10th Int. Conf. on Wind Engineering: Wind Engineering into the 21st Century, Balkema, Rotterdam, The Netherlands, Der Kiureghian, A Structural response to stationary excitation. J. Engrg. Mech. Div., 106 6, Diana, G., Bruni, S., Cigada, A., and Zappa, E Complex aerodynamic admittance function role in buffeting response of a bridge deck. J. Wind. Eng. Ind. Aerodyn., 90, Diana, G., Cheli, F., Zasso, A., and Bocciolone, M Suspension bridge response to turbulent wind: Comparison of new numerical simulation method results with full scale data. Proc., 10th Int. Conf. on Wind Engineering: Wind Engineering into the 21st Century, Balkema, Rotterdam, The Netherlands, Ge, Y. J., and Tanaka, H Aerodynamic stability of long-span suspension bridges under erection. J. Struct. Eng., , Irwin, P. A The role of wind tunnel modeling in the prediction of wind effects on bridges. Bridge aerodynamics, Balkema, Rotterdam, The Netherlands, Jones, N. P., Scanlan, R. H., Jain, A., and Katsuchi, H Advances and challenges in the prediction of long-span bridge response to wind. Bridge aerodynamics, Balkema, Rotterdam, The Netherlands, Scanlan, R. H. 1978a. The action of flexible bridges under wind. I: Flutter theory. J. Sound Vib., 60 2, Scanlan, R. H. 1978b. The action of flexible bridges under wind. II: Buffeting theory. J. Sound Vib., 60 2, Scanlan, R. H Problematics in formulation of wind-force models for bridge decks. J. Eng. Mech., 119 7, Soong, T. T., and Grigoriu, M Random vibration of mechanical and structural systems, Prentice-Hall, Englewood Cliffs, N.J. Xu, Y. L., Sun, D. K., Ko, J. M., and Lin, J. H Fully coupled buffeting analysis of Tsing Ma suspension bridge. J. Wind. Eng. Ind. Aerodyn., 85 1, Yamada, H., Miyata, T., Minh, N. N., and Katsuchi, H Complex flutter-mode analysis for coupled gust response of the Akashi Kaikyo Bridge model. Proc., 10th Int. Conf. on Wind Engineering: Wind Engineering into the 21st Century, Balkema, Rotterdam, The Netherlands, JOURNAL OF STRUCTURAL ENGINEERING ASCE / DECEMBER 2006 / 2017