New developments in bridge flutter analysis

Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri Proceedings of the Institution of Civil Engineers Structures and Buildings 165 March 2012 Issue SB3 Pages 139–159 http://dx.doi.org/10.1680/stbu.2012.165.3.139 Paper 900008 Received 20/01/2009 Accepted 01/04/2011 Keywords: bridges/dynamics/wind loading & aerodynamics ICE Publishing: All rights reserved New developments in bridge flutter analysis 1 j Claudio Mannini PhD Post-Doctoral Research Fellow, Inter-university Research Centre on Building Aerodynamics and Wind Engineering, Department of Civil and Environmental Engineering, University of Florence, Italy 2 Gianni Bartoli PhD j Associate Professor and Deputy Head of Inter-university Research Centre on Building Aerodynamics and Wind Engineering, 1 j 2 j Department of Civil and Environmental Engineering, University of Florence, Italy 3 Claudio Borri Prof. Ing. Dr-Ing. H.c. Mult j Full Professor and Head of Inter-university Research Centre on Building Aerodynamics and Wind Engineering, Department of Civil and Environmental Engineering, University of Florence, Italy 3 j The first part of this paper is devoted to an approximate approach to flutter, which is attained through simplification of the flutter equations. The critical wind speed and the flutter frequency can be calculated with the proposed formulas by employing only three flutter derivatives instead of the usual eight coefficients. This approach may be seen as an easy engineering tool for a better tailoring of bridge structures at early design stages. In addition, the simplicity of the equations allows better understanding of the mechanism of flutter instability and the role played by structural parameters such as damping. In particular, an explanation is provided for soft- and hard-type flutter. The second part of the paper outlines a model to take into account the uncertainty in the measurement of self-excited forces in a flutter analysis. Ad hoc wind tunnel tests allowed determination of the statistical properties of the measured flutter derivatives. These coefficients are treated as independent normally distributed random variables, and Monte Carlo simulations are performed in order to determine the probability distribution function of the critical wind speed. The paper concludes with an example of application of the proposed probabilistic flutter assessment method. 1. Introduction Modern long-span bridges are more and more sensitive to wind loads and aeroelastic phenomena, due to challenging designs involving high-performance materials leading to lighter structures and lower vibration frequencies. In particular, an adequate and reliable safety margin with respect to the critical wind speed leading to the aeroelastic instability known as flutter, which induces catastrophic oscillations and even collapse of the structure, has to be guaranteed. Classical flutter is a self-excited phenomenon due to the aeroelastic coupling of vertical bending and torsional modes that introduces energy into the system, leading to divergent or largeamplitude limit-cycle oscillations. Torsional flutter (or torsional galloping) is also relevant to bridge structures, wherein negative damping in a torsional mode can be attained without any coupling with other modes. Both phenomena are usually approached using semi-empirical models (Caracoglia and Jones, 2003; Costa and Borri, 2006; Scanlan, 1978a, 1978b; Scanlan and Tomko, 1971; Simiu and Scanlan, 1996) in which some aerodynamic coefficients (the so-called flutter derivatives) have to be experimentally determined in a wind tunnel. Given the strong dependence of the calculated flutter wind speed on these coefficients, the experimental phase and the identification procedure are extremely important (Bartoli et al., 2009). The flutter derivatives are, among other functions and coefficients, also necessary for accurate estimation of the structural response to turbulent wind (Scanlan, 1978b). Two different aspects related to the analysis of the flutter phenomenon are considered in this paper. Firstly, although it is important to compute in the most accurate way possible the flutter critical wind speed and the response to turbulent wind, the authors have attempted to simplify the flutter problem, searching for approximate formulas for flutter prediction to be employed at early design stages and implemented for the improvement of codes and standards (Mannini et al., 2007). Approximate approaches also help towards better understanding of the mechanism leading to flutter instability and consequently allow better tailoring of the structural design in order to avoid flutter. Several attempts to obtain simplified models of flutter assessment 139 Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri have been reported (Bartoli and Mannini, 2008; Bartoli and Righi, 2006; Chen, 2007; Dyrbye and Hansen, 1997; Frandsen, 1966; Nakamura, 1978; Mannini, 2006; Øiseth et al., 2010; Vairo, 2010), demonstrating the relevance of this issue. A comparison of the present approach and the methods proposed by Nakamura (1978) and Chen (2007) will be discussed in the next section. Some researchers (Bartoli and Righi, 2006; Dyrbye and Hansen, 1997; Frandsen, 1966) tried to condense the aeroelastic performance of a bridge deck section in one empirical coefficient, the aerodynamic stability performance index. In fact, the flutter critical wind speed can be calculated first for a dynamically equivalent theoretical flat plate and then corrected through this coefficient, which is usually less than one. This method is appealing for its simplicity but some drawbacks have already been highlighted (Bartoli and Mannini, 2008; Mannini, 2006). Vairo (2010) set up an analytical approach based on a simplified variational formulation for the dynamic problem of the wind–structure interaction mechanism leading to flutter instability in the case of cable-stayed bridges. However, the most common simplified method of flutter assessment is the quasisteady theory. The limits of this approach have been discussed by Bisplinghoff et al. (1996) and Bartoli and Mannini (2008), while its absolute inadequacy to predict torsional flutter instability was highlighted by Nakamura and Mizota (1975) and Nakamura (1979). Øiseth et al. (2010) proposed a modified quasi-steady approach, based on the linear or quadratic fit of experimentally measured flutter derivatives, as an engineering tool to estimate the flutter critical condition. coefficient of variation of 0.20 was assumed. The same simple model for the uncertainty affecting flutter derivatives was assumed by Cheng et al. (2005) who, in a reliability analysis of the Jing Yin Bridge, China, included a large number of random variables and then performed a sensitivity study of the reliability index with respect to the mean and standard deviation values of these variables. However, in no research work has the actual stochastic nature of the flutter derivatives been studied and no probabilistic model has been proposed to determine the probability distribution of the critical flutter wind speed, given the uncertainty in wind tunnel measurements. Therefore, in the third section of this paper, a method is proposed in order to account for aerodynamic input uncertainty in flutter analysis. A second issue dealt with in this paper is related to the experimental evidence that flutter derivatives involve uncertainties (i.e. their measurement presents a significant dispersion), in particular in the case of free-vibration setups (Mannini, 2006; Righi, 2003). The literature indicates several attempts to account for the stochastic effect on flutter boundaries due to oncoming flow turbulence (Bartoli and Righi, 2006; Lin, 1996), but in only a few cases and in a very approximate way has the uncertainty in flutter derivatives been considered in the reliability analysis of bridge structures. Ostenfeld-Rosenthal et al. (1992) considered a log-normally distributed random variable to account for uncertainty in structural damping and a Gaussian random variable for the uncertainty factor related to the conversion from model to full scale. In addition, the critical wind speed directly measured in a wind tunnel was assumed to be a Gaussian random variable. Ge et al. (2000) used an empirical formula to calculate the flutter critical wind speed, considering uncertainty in the bridge deck mass, mass moment of inertia and structural damping. A normally distributed conversion factor from model to prototype was also assumed. Pourzeynali and Datta (2002) considered the resistance of a bridge structure as the product of the flutter critical wind speed, log-normally distributed, and three independent log-normally distributed random variables accounting for the uncertainty in structural damping, in mathematical modelling and in the flutter derivatives. In particular, for the latter an almost arbitrary 140 2. Simplified approach to flutter instability 2.1 Mechanical model In flutter analysis, normally only the onset instability wind speed is sought for the design of bridge structures. Therefore, under the assumption of small oscillations perturbing the flow, the structure can be modelled as a two degrees of freedom (dof) linear oscillator: 1: 2: h i _ þ ø2 h(t) m h€(t) þ 2æ h ø h h(t) h
 _ Æ(t), Æ(t), _ K ¼ Lse h(t), h(t),   € (t) þ 2æÆ øÆ Æ(t) I Æ _ þ ø2Æ Æ(t)
 _ ¼ M se h(t), h(t), Æ(t), Æ(t), _ K where h and Æ are the heaving displacement and the pitching rotation (Figure 1), m and I the mass and the mass moment of B U h M α L Figure 1. Reference scheme for displacements and self-excited forces Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri inertia per unit length, ø h and øÆ are the circular eigenfrequencies, æ h and æÆ represent structural damping in heaving and pitching modes respectively and Lse and Mse are the self-excited lift and moment per unit length. K ¼ Bø/U is the reduced frequency of oscillation, where B is the deck width, ø the circular frequency of oscillation at flutter and U the undisturbed mean flow speed; the dots denote derivatives with respect to time t. Classically, the self-excited forces can be assumed to be linear functions of structural displacements and velocities, parametrically dependent on the non-dimensional reduced frequency of oscillation (Scanlan, 1978a, 1978b; Scanlan and Tomko, 1971; Simiu and Scanlan, 1996): 3 and 4 has been found to be questionable in some instances (Diana et al., 2008; Mannini et al., 2010a; Noda et al., 2003), since a non-negligible dependence of flutter derivatives on the amplitude of oscillation was observed.  _ BÆ(t) _ h(t) þ KH
2 (K) U U  h(t) 2
(K)Æ(t) þ K H (K) þ K2 H
3 4 B Lse ¼ qB KH
1 (K) 3: By assuming coupled harmonic oscillations at frequency ø and imposing the system complex determinant to vanish, one obtains fourth-order and third-order polynomial equations with respect to the non-dimensional frequency of oscillation Y ¼ ø/ø h (the socalled real and imaginary flutter equations (e.g. Dyrbye and Hansen, 1997)). The polynomial coefficients are non-linear functions of the reduced frequency of oscillation K. The couple (Y, Kc ), which identifies the critical condition, can be obtained as the intersection of the solutions of the real and imaginary flutter equations. Then the critical wind speed can be easily calculated from: 5:  _ BÆ(t) _ h(t) þ KA
2 (K) U U  h(t) 2
2
þ K A3 (K)Æ(t) þ K A4 (K) B M se ¼ qB2 KA
1 (K) 4: where q ¼ 0.5rU 2 is the mean dynamic pressure, r is the air density and the coefficients H
i and A
i are the flutter derivatives, which are functions of the reduced frequency of oscillation K as well as the mean angle of attack (Diana et al., 2004; Mannini, 2006; Mannini and Bartoli, 2008a). This simple 2-dof model can be extended to multi-mode analysis (Chen and Kareem, 2006; Chen et al., 2000; Jain et al., 1996; Katsuchi et al., 1999; Scanlan, 1978a, 1978b) of the bridge structure. Nevertheless, in most cases the flutter critical wind speed can be accurately calculated by considering two modes only (Bartoli and Mannini, 2005a; Chen et al., 2000). A complete flutter approach also requires inclusion of the self-excited drag force and the contribution of the along-wind (sway) motion (Chen et al., 2000; Jain et al., 1996; Katsuchi et al., 1999; Scanlan, 1978a, 1978b), so that the flutter derivatives to be determined in the wind tunnel total 18 instead of eight. In most cases, the contribution of drag and sway motion were found to be negligible (Bartoli and Mannini, 2005a; Chen and Kareem, 2006; Chen et al., 2000; Øiseth et al., 2010), although in a few instances a significant effect on the flutter critical condition was encountered (Katsuchi et al., 1999; Vairo, 2010). Chen and Kareem (2006) explained that the additional flutter derivatives are non-negligible only if the drag is large and the self-excited lift and moment are small, so that the contribution of the latter to the aerodynamic damping builds up slowly with increasing wind speed. It is worth remarking that the classical linear model of Equations U c ¼ Bø h Y Kc Alternatively, it is possible to define the dissipative forces in the left-hand side of Equations 1 and 2 as imaginary stiffness terms (Fung, 1993) by introducing the rate-independent damping coefficients g h and gÆ : h i m €h(t) þ (1 þ ig h )ø2h h(t) 6: 7:
 _ Æ(t), ia(t), K ih ¼ Lse h(t), h(t),   € (t) þ (1 þ ig Æ )ø2Æ Æ(t) I Æ
 _ i ¼ M se h(t), iah(t), Æ(t), Æ(t), _ K where i ¼ (1)1=2 denotes the imaginary unit. The rate-independent damping coefficients can be related to the ratio-to-critical damping coefficients by the following expressions: 8: g h ¼ 2æ h ø øh 9: g Æ ¼ 2æÆ ø øÆ The previously mentioned flutter equations can now be written as: 141 Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri    ì2 2 2 1 2 2 ð1  g h g Æ Þ 2 r Æ X  r Æ ì 1 þ 2 ªø ªø þ 12:  ì


 þ r 2 ì
H
 g H
 X A  g A Æ h 4 1 3 2 Æ ª2ø
þ r2Æ ì2 þ ìA3
þ r2Æ ì H
4 þ H
4 A3 10:




 H
1 A2  H 3 A4 þ H 2 A 1 ¼ 0 ð g h þ gÆ Þ þ     ì2 2 2 gh 2 2 r X  r ì g þ Æ Æ ª2ø ª2ø Æ #

 ì

2
 X A2 þ g h A
3 þ rÆ ì H 1 þ g Æ H 4 2 ªø


 H
3 A1  H 2 A4 ¼ 0 where  1=2 1 I rÆ ¼ B m is the non-dimensional radius of gyration, ì¼ BøÆ K c X 1=2 2.2 Approximate formulas It is possible to remark that by assuming 1  g h g Æ ffi 1, Equation 10 can be written as: 2




þ ìA
2 þ r Æ ì H 1 þ H 4 A2 þ H 1 A3 11: Uc ¼ 13:   X 2 2
ìA3  rÆ ì (X  1) 1  2 ªø
X  r2Æ ì H
4 (X  1) þ g h ìA
2 þ gÆ ª2ø r2Æ ì H
1 ª2ø þ H
4 A
3  H
1 A
2  H
3 A
4 þ H 2
A
1    X ffi ìA
3  r2Æ ì2 (X  1) 1  2 ¼ 0 ªø The approximation in Equation 13 holds for all the dynamic and aerodynamic data of real bridge structures collected by the authors in a database (Bartoli and Mannini, 2005b, 2008; Mannini, 2006), unless X ffi ª2ø (i.e. ø ffi ø h ), a condition which occurs only if the still air frequency ratio is very close to one. A simplified equation is therefore obtained for the frequency parameter: 2m rB2 14: X ¼1þ A
3 r2Æ ì is the mass ratio of the deck section, øÆ ªø ¼ øh is the still-air frequency ratio and X ¼ ø2Æ =ø2 Furthermore, Equations 10 and 11 can be linearly combined in order to eliminate the term proportional to X 2 and all the terms that contain a second power of the damping coefficients can be neglected. Several terms in the resulting equation are found to be negligible with respect to the others, once the range of variability of the dynamic and aerodynamic parameters has been identified with reference to the previously mentioned bridge database (Bartoli and Mannini, 2005b, 2008; Mannini, 2006). Therefore, the following simplified equation can be obtained: is a non-dimensional frequency parameter. The advantage of this approach is that the flutter equations simplify to second-order polynomial equations of the frequency parameter X and hence manipulations are easier. Given the couple (Xc , Kc ) which satisfies Equations 10 and 11, one can calculate the flutter critical wind speed as: 142 15: X ¼ ª2ø r2Æ ì( g h þ gÆ ) þ A
3 ( g h þ gÆ )  A
2  r2Æ H
1 r2Æ ì(ª2ø g h þ gÆ ) þ gÆ A
3  A
2  ª2ø r2Æ H
1 By comparison with Equation 14, one obtains: Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri 2 2
2

gÆ (A
3 ) þ g Æ r Æ ìA3 (2  ªø )  A2 A3
2
2  ª2ø r2Æ H
1 A3 þ r Æ ìA2 (ªø  1)  r4Æ ì2 gÆ (ª2ø  1) ¼ 0 16: Finally, observing that the first two terms of Equation 16 are negligible with respect to the others, the following simplified equations can be obtained:
2 2

2
2 A
2 A3 þ ªø rÆ H 1 A3  r Æ ìA2 (ªø  1) 17: þ r4Æ ì2 gÆ (ª2ø  1) ¼ 0 Equation 17 allows determination of the flutter critical reduced frequency of oscillation, which appears as the argument of the flutter derivatives. Then, through Equation 14, the non-dimensional frequency parameter X can be calculated and, through Equation 12, the critical wind speed is finally obtained. More details about the simplification procedure can be found in previous publications (Bartoli and Mannini, 2005b, 2008; Mannini, 2006) in which Equations 15 and 16 had been taken as final approximate formulas. Compared with these previous formulations, the first two terms of Equation 16 were discovered to be always negligible in practice, while the equation for the frequency parameter (Equation 15) was substituted by a simpler and more meaningful relation (Equation 14) that involves only the flutter derivative A
3 instead of H
1 , A
2 and A
3 : Apart from the convenient analytical form of the proposed equations, it is important to stress the fact that only three out of eight flutter derivatives are retained. These functions are the so-called ‘uncoupled’ flutter derivatives, which can be measured in a wind tunnel with 1-dof experimental setups. Clearly, Equations 14 and 17 represent a remarkable simplification of the flutter problem. From analysis of Equation 14 one can notice that the mechanism with which the torsional frequency decreases up to the flutter frequency is very simple and analogous to the case of 1-dof systems, that is, depending only on the flutter derivative A3
, and is unaffected by structural damping. In fact, from Equation 14 it is possible to write: 18: ø ¼ øÆ  rB4
1þ A 2I 3 1=2 Conversely, the analysis of Equation 17 is less immediate but it can be remarked that structural damping can play a significant role in the equation for the critical reduced frequency of oscillation through one term depending only on the structural properties of the bridge. These observations are in agreement with the results of Chen and Kareem (2003) who found that structural damping does not significantly influence frequency, aerodynamic damping or complex mode shape and the aeroelastic modal damping can be simply estimated as the sum of structural modal damping and aerodynamic damping estimated with zero structural damping. Moreover, Equation 17 shows that only structural damping of the torsional mode has an influence on the flutter mechanism. This is reasonable as, usually, it is the evolution under wind of the torsional mode (torsional branch) that becomes unstable; in other words, at the critical wind speed it is the eigenvalue relative to torsion that exhibits a positive real part. The assumption that the heaving branch is stable was also stated by Nakamura (1978) in obtaining a simplified formula of flutter assessment. Nevertheless, Matsumoto et al. (1999, 2002, 2008) observed that, in a few particular cases, the flutter derivative A2
assumes sufficiently large negative values to stabilise the torsional branch. In this instance it is the heaving branch that gives rise to the flutter instability. This issue will be discussed later on in the paper. As is clear from the procedure of simplification of the flutter equations, the approximate Equations 14 and 17 do not hold for frequency ratios very close to unity, in the same way as Selberg’s (Selberg, 1961) and Rocard’s (Frandsen, 1966) formulas for a flat plate. Nevertheless, it seems that a frequency ratio of about 1.3, or sometimes even less, is sufficient to obtain an acceptable degree of approximation. Frequency ratios very close to unity are fairly uncommon and usually characterise either super-long-span bridges or very unconventional structures that are not expected to be analysed with simplified methods, requiring instead careful experimental campaigns from the very beginning of the design procedure. If damping is neglected, the proposed model reduces to that proposed by Nakamura (1978), although it was obtained in a completely different way (Bartoli and Mannini, 2008; Mannini, 2006). The simplified closed-form solution of the flutter equations proposed by Chen (2007) accounts for the structural damping contribution and is also based on the assumption of wellseparated frequencies. Chen’s approach retains only four flutter


derivatives, namely H
3 , A1 , A2 and A3 : The fact that the flutter derivatives considered here are different from those emphasised in that work should not be seen as a contradiction in view of existing flutter derivative inter-relations (Matsumoto, 1996; Scanlan et al., 1997). In particular, the fact that the proposed formulas

depend only on the coefficients H
1 , A2 and A3 does not mean that these are the most important derivatives for the instability mechanism and that the others can be neglected. On the contrary, the apparent redundancy of the classical model of self-excited forces (Equations 3 and 4) just allows expression of the critical condition with three aerodynamic parameters instead of eight. An important feature of the model is that the present equation for the critical reduced frequency of oscillation does account for the 143 Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri contribution of structural damping is an important feature of the model. In fact, as noted by Chen and Kareem (2003), while the role of structural damping on the onset of instability is negligible in the case of ‘hard-type flutter’, it is significant in the case of ‘soft-type flutter’. In the latter instance, negative damping builds up slowly with increasing wind speed, so that a small translation of the curves of total damping due to an increase of the structural damping is able to induce a significant increment of the critical wind speed. By contrast, the change of sign of the total damping is abrupt for hard-type flutter and therefore the instability condition is only weakly sensitive to small translations of its curve. Only in the case of soft-type flutter devices such as tuned mass dampers can be effective in controlling flutter instability (Chen and Kareem, 2003) and the effect of large self-excited drag forces can be non-negligible (Chen and Kareem, 2006). This issue will be further analysed later in the paper. tests were performed considering the flutter derivatives of a theoretical flat plate (Fung, 1993; Theodorsen, 1934), which is always an important benchmark for bridge aeroelasticity, and a rectangular cylinder with a chord to thickness ratio B/H ¼ 12.5 (specimen R12.5, the bluffer one among those studied by Matsumoto (1996) not prone to torsional flutter). Finally, if damping is provided as a ratio-to-critical instead of a rate-independent coefficient, the procedure of calculation of the critical reduced frequency of oscillation through Equation 17 becomes iterative. Equation 9 with ø ¼ øÆ is used to obtain a first estimate of g Æ and then, once the critical flutter frequency has been calculated through Equation 14 or 18 , the rateindependent damping coefficient is updated and the flutter calculation is repeated. One iteration is normally enough to obtain convergence. 2.3 Model validation In order to validate Equations 14 and 17, the following procedure was adopted. A large number of structural parameters for various typologies of existing bridges (suspension, cable-stayed and footbridges) were collected and compared (Bartoli and Mannini, 2005b, 2008; Mannini, 2006). Since mass and moment of inertia enter in the flutter equations (Equations 10 and 11) in the form of the non-dimensional parameters ì, r2Æ ì and r2Æ ì2 , two bridge structures were identified as representative of opposite extreme dynamics and used as references in the following analyses: Tsurumi Fairway Bridge in Japan (ì ¼ 35:7 and rÆ ¼ 0:249) and Rio Guamà Bridge in Brazil (ì ¼ 178:6 and rÆ ¼ 0:353). Both are cable-stayed bridges but Rio Guamà Bridge is characterised by a concrete deck and consequently by a remarkable mass with respect to the chord B. In contrast, Tsurumi Fairway Bridge has a relatively light steel deck with a large chord length. In all the flutter calculations presented in the following, air density was always assumed to be r ¼ 1.25 kg/m3 : In order to draw as general as possible conclusions, it is important to consider a large number of dynamic and aerodynamic data. Assuming that flutter derivatives depend on reduced frequency of oscillation and cross-section geometry only, it is possible to combine the aerodynamics of a bridge with the dynamic properties of a completely different one, performing calculations on idealised structures. This is important: it allows one to employ all the reliable aerodynamic and dynamic data available for the calculations. In particular, the first validation 144 Figure 2 combines the previously discussed reference dynamics and aerodynamics, and compares the rigorous solution of the flutter equations (Equations 10 and 11) with those given by the approximate formulas (Equations 14 and 17) in terms of critical reduced wind speed URc ¼ 2ð/Kc for different values of the frequency ratio ªø and structural damping. It is clear that the approximation offered by the simplified method is good unless the frequency ratio is very close to unity. It is also apparent in Figure 2(d) that when the effect of damping is significant, the simplified equations are able to correctly account for it. A similar analysis was performed for the case of a deck crosssection prone to torsional flutter. In particular, the flutter derivatives measured by Matsumoto (1996) for a rectangular cylinder with a chord to thickness ratio B/H ¼ 5.0 (specimen R5) were considered, as this geometry is considered a benchmark test case for bridge aerodynamics and aeroelasticity (Bartoli et al., 2008). The results are reported in Figure 3, along with the solution of the 1-dof problem. It is worth noting that the 2-dof and 1-dof approaches usually give close results, unless the frequency separation is small. Nevertheless, Chen et al. (2000) and Chen and Kareem (2006) observed that the coupling of heaving and pitching motions generally reduces the critical wind speed in the case of torsional flutter. The agreement between exact and approximate solutions is again very good and the simplified equations are able to take into account the unfavourable effect of coupling between bending and torsion and the role played by damping (Figure 3(b)). As further validation, various case studies were taken into account for classical and torsional flutter instabilities (Tables 1– 4). In particular, cases 5 and 6 in Tables 1 and 2 refer respectively to Akashi Kaikyo Bridge (Katsuchi et al., 1999) in Japan and Tsurumi Fairway Bridge in Japan (Singh et al., 1995); cases 1 and 9 refer to the Original Tacoma Narrows Bridge, USA (Scanlan and Tomko, 1971) and Golden Gate Bridge, USA (Simiu and Scanlan, 1996) for both dynamic and aerodynamic properties. All the other case studies are idealised. The flutter derivatives of rectangular cylinders with chord to thickness ratios of 20 (R20 (Matsumoto, 1996)) and 10 (R10 (Matsumoto, 1996)) and a rectangular cylinder with semi-circular fairings and a chord to thickness ratio of 14.3 (R14.3F (Chowdhuri and Sarkar, 2004)) were also studied. It is worth noting in Tables 3 and 4 that, according to the flutter classification outlined by Matsumoto et al. (2002), R5, the Tacoma and the Golden Gate sections are prone to the so-called low-speed torsional flutter, while R10 is susceptible to high-speed torsional flutter. Concerning structural properties, case 2 of Tables 1 and 2 and Structures and Buildings Volume 165 Issue SB3 45 Flat plate – µ ⫽ 35·7, rα ⫽ 0·249 16 40 14 35 12 30 10 URc URc 18 New developments in bridge flutter analysis Mannini, Bartoli and Borri 8 ζh ⫽ ζα ⫽ 0%; exact solution 25 20 6 ζh ⫽ ζα ⫽ 1%; exact solution 15 4 ζh ⫽ ζα ⫽ 0%; approximate solution 10 2 ζh ⫽ ζα ⫽ 1%; approximate solution 5 0 0 1 14 2 γω 3 4 5 1 2 γω 3 4 (a) (b) R12.5 – µ ⫽ 35·7, rα ⫽ 0·249 R12.5 – µ ⫽ 178·6, rα ⫽ 0·353 5 20 12 10 15 8 URc URc Flat plate – µ ⫽ 178·6, rα ⫽ 0·353 6 10 4 5 2 0 1 2 γω 3 4 5 (c) 0 1 2 γω 3 4 5 (d) Figure 2. Comparison of approximate and exact solutions of the flutter equations (URc ¼ 2ð/Kc is the flutter critical reduced wind speed) for different values of the frequency ratio ªø and two values of the ratio-to-critical damping coefficients in the case of classical flutter instability case 5 of Tables 3 and 4 refer to the 1992-proposed design of Messina Strait Bridge, Italy; case studies 4 and 9 of Tables 1 and 2 refer to Bosporus I Strait Bridge, Turkey; case study 7 of Tables 1 and 2 refers to Indiano Bridge, Italy; case study 10 of Tables 1 and 2 refers to Normandy Bridge, France. The interested reader can find more details about the dynamic properties of these bridges in the literature (Bartoli and Mannini, 2005b, 2008; Mannini, 2006). It is apparent in the tables that the degree of approximation offered by the simplified formulas is good, often limited to a few percent and in most cases below 10%. Only in cases 2, 6 and 7 (Tables 1 and 2) are the errors larger, but still well below 20%. For case 2, the discrepancy was expected as the frequency ratio is small, namely ªø ¼ 1:32 (the dynamic parameters of the 1992-proposed design of Messina Strait Bridge are considered in this case). In contrast, test cases 6 and 7 refer to the flutter derivatives of Tsurumi Fairway Bridge. As shown in Figure 4, for this fairly streamlined cross-section, the values assumed by the coefficient H
4 are surprisingly high, thus rendering the proposed formulas slightly less accurate. In fact, the absolute value of the coefficient is in this case much larger not only than the one predicted by Theodorsen’s theory for a flat plate (Fung, 1993; Theodorsen, 1934) but also for most bridge deck sections, which usually show values close to zero. This is the reason why this coefficient was neglected along with A
4 in several formulations (e.g. Scanlan, 1978a, 1978b; Scanlan and Tomko, 1971). Tables 2 and 4 show that the approximate formula for the critical frequency is very accurate. Also, agreement with the rigorous 2dof results is extremely good in the case of structures prone to torsional flutter (both for low-speed and high-speed torsional flutter; Table 4) and generally significantly better than the results offered by the 1-dof approach. Moreover, comparison with the results reported by Bartoli and Mannini (2008) demonstrates that 145 Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri R5 – µ ⫽ 35·7, rα ⫽ 0·249 the additional simplification of the formulas introduced here does not imply any significant loss of accuracy. 5 URc 4 Figure 5 shows the pattern of the function on the right-hand side of Equation 17 and its zero crossing as compared with the exact solution of the flutter equations in the case of Akashi Kaikyo Bridge (case 5 in Tables 1 and 2) and the Original Tacoma Narrows Bridge (case 1 in Tables 3 and 4). ζh ⫽ ζα ⫽ 0%; exact solution 3 ζh ⫽ ζα ⫽ 1%; exact solution ζh ⫽ ζα ⫽ 0%; approximate solution 2 ζh ⫽ ζα ⫽ 1%; approximate solution ζh ⫽ ζα ⫽ 0%; 1-dof solution 1 ζh ⫽ ζα ⫽ 1%; 1-dof solution 0 1 2 4 3 γω (a) 5 R5 – µ ⫽ 178·6, rα ⫽ 0·353 6 5 URc 4 3 2 1 0 1 2 γω (b) 4 3 5 Figure 3. Comparison of approximate and exact 1-dof and 2-dof solutions of the flutter equations (URc ¼ 2ð/Kc is the flutter critical reduced wind speed) for different values of the frequency ratio ªø and two values of the ratio-to-critical damping coefficients in the case of torsional flutter instability Case Aerodynamics 1 2 3 4 5 6 7 8 9 10 Flat plate Flat plate R12.5 R12.5 Akashi Kaikyo Bridge Tsurumi Fairway Bridge Tsurumi Fairway Bridge R14.3F R14.3F R20 Table 1. Case studies (classical flutter) 146 ì rÆ ªø 35.7 24.1 178.6 27.7 55.6 35.7 55.5 47.2 27.7 38.7 0.249 0.374 0.353 0.357 0.422 0.249 0.250 0.539 0.357 0.286 2.38 1.32 1.96 2.29 2.34 2.38 2.06 1.54 2.29 2.27 The case of wind tunnel section models, to which Equations 14 and 17 also apply, is very particular. As an example, a common trapezoidal box girder with lateral cantilevers (Figure 6), similar to Sunshine Skyway Bridge in Florida, tested in the CRIACIV (Inter-university Research Centre on Buildings Aerodynamics and Wind Engineering) wind tunnel, was considered (Mannini, 2006; Mannini et al., 2010b). Two different configurations were selected: the second one was simply obtained from the first by adding eccentric masses to the suspension system (for configuration Dyn. 1 B ¼ 0:45 m, rÆ ¼ 0:294 and ì ¼ 45:0; for Dyn. 2 B ¼ 0:45 m, rÆ ¼ 0:426 and ì ¼ 52:0). Flutter derivatives were measured on configuration Dyn. 1. It is worth noting that, from a structural dynamic point of view, these two test cases are within the range defined by the previously mentioned reference dynamics (i.e. those of Tsurumi Fairway Bridge and Rio Guamà Bridge) in spite of the fact that the section model was not dynamically scaled with respect to any real bridge but just designed to measure flutter derivatives (Mannini, 2006). Figure 7 shows a comparison of the approximate and exact solutions in terms of the dimensional critical wind speed for several values of the frequency ratio and two ratio-to-critical damping coefficients. The agreement between the different sets of results is good up to small frequency separations. In some particular cases, it is not the torsional branch that is unstable, but coupled flutter arises in the heaving-branch (Matsumoto et al., 1999, 2002, 2008). For instance, this is the case for a rectangular cylinder with chord to thickness ratio B/H ¼ 20 and vertical plates at mid-chord (R20-VP (Matsumoto et al., 1999, æ h : % æÆ : % 0.5 0.6 0.8 0.5 0.5 0.5 0.2 0.5 0.5 0.2 0.5 0.7 0.8 0.5 0.3 0.5 0.2 0.5 0.5 0.5 Structures and Buildings Volume 165 Issue SB3 Case New developments in bridge flutter analysis Mannini, Bartoli and Borri Exact solution 1 2 3 4 5 6 7 8 9 10 Approximate solution fc : Hz URc Uc : m/s fc : Hz ˜fc : % URc ˜URc : % Uc : m/s ˜Uc : % 0.315 0.071 0.562 0.308 0.138 0.377 0.880 0.180 0.284 0.338 10.75 6.72 16.80 7.64 16.14 8.34 11.04 11.41 10.16 11.21 128.7 28.9 134.2 66.0 79.2 119.5 217.6 24.6 80.9 90.2 0.315 0.071 0.563 0.314 0.138 0.352 0.836 0.179 0.283 0.340 +0.1 +0.5 +0.1 +1.7 0.3 6.7 5.0 0.1 0.6 +0.7 10.43 5.51 16.83 7.01 15.91 9.89 12.72 10.59 9.35 10.79 3.0 18.0 +0.2 8.2 1.5 +18.5 +15.3 7.2 8.0 3.7 124.9 23.8 134.5 61.6 77.2 132.2 238.3 22.8 74.0 87.4 2.9 17.6 +0.2 6.6 1.7 +10.6 +9.5 7.3 8.5 3.1 Table 2. Results for the case studies in Table 1; fc is the coupling frequency, URc is reduced flutter wind speed and Uc is flutter wind speed Case Aerodynamics 1 2 3 4 5 6 7 8 9 10 Tacoma Narrows Bridge R5 R5 R5 R5 R10 R10 R10 Golden Gate Bridge Golden Gate Bridge ì rÆ ªø 47.2 47.2 35.7 178.6 24.1 47.2 35.7 178.6 94.2 35.7 0.539 0.539 0.249 0.353 0.374 0.539 0.249 0.353 0.369 0.249 1.54 1.54 2.38 1.96 1.32 1.54 2.38 1.96 2.20 2.38 æ h : % æÆ : % 0.5 0.5 0.5 0.8 0.6 0.5 0.5 0.8 0.5 0.5 0.5 0.5 0.5 0.8 0.7 0.5 0.5 0.8 0.5 0.5 Table 3. Case studies (torsional flutter) Case 1-dof solution fc : Hz 1 2 3 4 5 6 7 8 9 10 URc 0.200 4.81 0.189 5.05 0.376 4.64 0.626 5.73 0.066 4.74 0.182 9.89 0.334 9.15 0.602 11.21 0.188 4.31 0.448 3.46 2-dof exact solution Approximate solution Uc : m/s fc : Hz URc Uc : m/s fc : Hz ˜fc : % URc ˜URc : % Uc : m/s ˜Uc : % 11.5 11.5 66.3 50.9 19.0 21.6 116.3 95.9 22.2 58.9 0.200 0.190 0.377 0.626 0.067 0.188 0.372 0.610 0.188 0.452 4.81 4.41 4.08 5.56 2.76 7.65 7.21 10.16 4.23 3.28 11.5 10.0 58.5 49.4 11.2 17.3 101.9 87.9 21.8 56.4 0.200 0.190 0.385 0.626 0.070 0.189 0.375 0.611 0.188 0.453 0.0 +0.4 +2.0 +0.1 +3.3 +0.1 +0.9 +0.2 +0.0 +0.1 4.81 4.35 3.94 5.52 2.90 7.46 6.99 10.00 4.10 3.10 0.0 1.4 3.2 0.7 +5.0 2.5 3.0 1.6 3.1 5.4 11.5 9.9 57.7 49.1 12.2 16.9 99.8 86.7 21.2 53.4 0.0 1.0 1.3 0.6 +8.5 2.3 2.1 1.4 3.0 5.3 Table 4. Results for the case studies in Table 3 (differences are calculated with respect to the 2-dof exact solution) 147 Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri 10 2 Exact Approximate 8 0 6 4 f (UR) H *4 ⫺2 ⫺4 2 0 ⫺6 Flatplate theory Tsurumi Fairway Bridge ⫺8 ⫺2 ⫺4 ⫺10 0 5 10 15 ⫺6 UR Figure 4. Comparison of theoretical values of the flutter derivative H
4 for a flat plate (Fung, 1993; Theodorsen, 1934) and measured values for the Tsurumi Fairway Bridge deck section (Singh et al., 1995) 5 0 15 10 UR 20 (a) 4 3 2 The case of vertical galloping is not taken into account here as this instability is rarely a problem for bridge designs. Moreover, a simple quasi-steady approach seems to be fairly adequate to estimate critical wind speed. 2.4 Analysis of the mechanism of instability The simplified formulas of Equations 14 and 17 can also be used to better understand the mechanism of flutter instability. For this purpose, all the terms in Equation 17 containing the aerodynamic stiffness coefficient A
3 can be moved to the right-hand side of the equation: 19: 2 A
2  rÆ ì g Æ ¼

 1 A
3 A2 þ ª2ø r2Æ H
1 2 2 ªø  1 rÆ ì 1 f (UR) 2002)), which shows large negative values of A
2 able to stabilise the torsional branch. A few case studies for this type of instability are presented in Tables 5 and 6. Despite the fact that they were considered for the torsional branch instability, the simplified formulas give reasonable results also in these cases. Nevertheless, by reducing the frequency ratio, the flutter equations do not allow any solution while the approximate approach does. The minimum frequency ratio for which a flutter solution is attained strongly depends on the bridge dynamic properties as is evident in Tables 5 and 6. 0 ⫺1 ⫺2 ⫺3 ⫺4 0 1 2 4 3 5 6 Figure 5. Graphic solution of the approximate Equation 17 for (a) Akashi Kaikyo Bridge and (b) Tacoma Narrows Bridge.

2 2 2

2 4 2 2 f (UR ) ¼ A
2 A3 + ªø r Æ H1 A3  r Æ ìA2 (ªø  1) + r Æ ì gÆ (ªø  1). The exact solution of the flutter equations is also reported equation can be interpreted as follows. The progressive reduction of the torsional frequency X  1 due to the aerodynamic coefficient A
3 is transformed into negative aerodynamic damping through a weighting factor depending on the flutter derivatives
H
1 and A2 , the frequency ratio and the non-dimensional radius of gyration. Instability is reached when this negative damping equals the positive structural damping and the direct aerodynamic damping in torsion that depends on A
2: Substituting Equation 14 into Equation 19 yields: Equation 19 can also be rearranged in the following manner: 20:  X 1

2 A2 þ ª2ø r2Æ H
A
1 2  rÆ ì g Æ ¼ 2 ªø  1 In the case of sections not prone to torsional flutter (A2
, 0), this 148 7 UR (b) 21: gÆ  A
2 ¼0 r2Æ ì Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri 450 150 150 150 2% Aluminium rib s ⫽ 8 mm Stiffners 70 65·5 4·6 4 2% 8 8 10 10 Sheet aluminium s ⫽ 0·5 mm 76·2 71·1 155·4 76·2 71·1 Figure 6. Section-model geometry (dimensions in millimetres) 70 ζh ⫽ ζα ⫽ 0%; exact solution Case Aerodynamics 40 1 2 3 4 R20-VP R20-VP R20-VP R20-VP 30 Table 5. Case studies (heaving-branch classical flutter) ζh ⫽ ζα ⫽ 1%; exact solution 60 ζh ⫽ ζα ⫽ 0%; approximate solution Uc: m/s 50 ζh ⫽ ζα ⫽ 1%; approximate solution ì rÆ ªø 35.7 35.7 178.6 178.6 0.249 0.249 0.353 0.353 2.38 1.65 1.96 1.20 æ h : % æÆ : % 0.5 0.5 0.8 0.8 0.5 0.5 0.8 0.8 20 where 10 0 0 1 2 70 γω (a) 3 4 5 22: 60 Uc: m/s 50 40 30 20 10 0 0 1 2 γω (b) 3 4 5 Figure 7. Comparison of approximate and exact solutions of the flutter equations (Uc is the dimensional critical wind speed) for different values of the frequency ratio ªø and two values of the ratio-to-critical damping coefficients in the case of the wind tunnel section model of Figure 6 (Mannini, 2006; Mannini et al., 2010b). (a) Dyn. 1 (B ¼ 0:45 m, rÆ ¼ 0:294, ì ¼ 45:0); (b) Dyn. 2 (B ¼ 0:45 m, rÆ ¼ 0:426, ì ¼ 52:0) A
2 ¼ A
2  1 A
2 A
3 ª2ø H
1 A
3  ª2ø  1 r2Æ ì ª2ø  1 ì Then the 2-dof flutter critical condition is written as for the 1-dof torsional flutter instability, by substituting the flutter derivative A
2 with the modified function A
2 : It is clear from Equation 21 that 2 A
2 =rÆ ì can be interpreted as aerodynamic damping of the unstable branch. In the case of a cross-section prone to torsional
flutter, the difference between A
2 and A2 tends to be small (Mannini and Bartoli, 2008b). The term proportional to H
1 A
3, which is equivalent to A
1 H
3 , gives the main destabilising contribution, in agreement with the conclusions of others (Chen, 2007; Matsumoto, 1996; Øiseth et al., 2010). Nevertheless, unless the non-dimensional mass moment of inertia r2Æ ì ¼ 2I=rB4 and
the frequency ratio are very large, the term proportional to A
2 A3 can also play a non-negligible destabilising role. This simple flutter formulation can give some indications to tailor a structure at early design stages and increase the flutter critical wind speed by modifying the structural dynamic properties of a bridge. In particular, it can help to understand to what extent the classical strategies are really efficient. Obviously an increase in torsional frequency (especially by increasing the torsional stiffness of the structure) has a positive effect because the dimensional critical wind speed increases once the reduced critical 149 Structures and Buildings Volume 165 Issue SB3 Case 1 2 3 4 New developments in bridge flutter analysis Mannini, Bartoli and Borri Exact solution Approximate solution fc : Hz URc Uc : m/s fc : Hz ˜fc : % URc ˜URc : % Uc : m/s ˜Uc : % 0.249 — 0.377 — 14.13 — 27.84 — 133.8 — 149.0 — 0.242 0.220 0.403 0.340 3.1 — +7.0 — 12.77 9.56 24.06 13.66 9.7 — 13.6 — 117.2 79.9 137.8 66.0 12.4 — 7.5 — Table 6. Results for the case studies in Table 5; fc is the coupling frequency, URc is reduced flutter wind speed and Uc is flutter wind speed The simplified formulas can also help to explain the reasons for the different types of flutter instability highlighted by Chen and Kareem (2003) and discussed in Section 2.2. It is worth noting that, once the bridge structural properties have been fixed, one can observe hard- or soft-type flutter depending on the aerodynamic properties, as clearly shown in Figure 8. In the same way, given the flutter derivatives, the type of instability can change depending on the structural properties of the bridge. From the analysis of Equations 21 and 22 it can be seen that a reduction in the absolute value of the flutter derivative, A
2 , or in the frequency ratio ªø , determines a decrease of the slope of A
2 near the critical condition, thus fostering soft-type flutter instability. Also, a diminution of the non-dimensional mass moment of inertia, r2Æ ìj has a similar effect. Instead, a variation of the non-dimensional mass, ì ¼ 2m=rB2 does not significantly change this slope. It is also worth noting that the structural damping coefficient g Æ in Equations 19–21 is directly scaled by r2Æ ì, so that the importance of damping is amplified if the mass moment of inertia of the deck section is large, in agreement with the results of Figures 2 and 3 (consider that r2Æ ì differs by a factor of about 10 between the Rio 150 0·4 R12.5, heaving branch R12.5, torsional branch Flat plate, heaving branch Flat plate, torsional branch 0·3 Total damping wind speed has been fixed (see Equation 12). However, particularly interesting is to understand how the reduced critical wind speed can be increased. From Equation 22 it is apparent that in the case of classical flutter instability an increase in deck mass per unit length m always has a favourable effect. Conversely, if A
2 takes small values and the frequency ratio is already fairly large, a further increment of the frequency separation does not produce any appreciable increase of the reduced critical wind speed (see Figures 2(c) and 2(d). In the same conditions, an increase of the mass moment of inertia I tends not to be effective. However, the reduced critical wind speed might be more sensitive to variations of the mass moment of inertia in the case of large values of the structural damping in torsion, as shown by Equation 21. In fact the discussed parameters are generally related (for instance, the frequency ratio depends on the ratio of the mass moment of inertia to the mass of the deck) and it is not possible to change them in a completely independent way. A detailed discussion of the effects of different structural dynamic parameters on the flutter instability mechanism can be found in Mannini and Bartoli (2008b). 0·2 Hard-type flutter 0·1 Soft-type flutter 0 ⫺0·1 ⫺0·2 0 5 15 10 20 25 U/(Bfα) Figure 8. Example of soft-type and hard-type flutter for the same structure (ì ¼ 178:6, rÆ ¼ 0:353, ªø ¼ 1:96, æ h ¼ æÆ ¼ 0) with different aerodynamic properties. The total modal damping (structural plus aerodynamic) is plotted against the normalised wind speed ( fÆ is the still-air torsional frequency) Gramà and Tsurumi Fairway bridges). Therefore, if the slope of the curve of the aerodynamic damping near the critical condition is small enough (soft-type flutter) and the non-dimensional mass moment of inertia is anyway large, then structural damping plays a non-negligible role on the mechanism of instability. It is clear from what has been said so far that a key parameter that contributes to trigger the type of instability is the flutter derivative, A
2 . In fact, Figure 9 shows that the rectangular cylinder R12.5 with respect to the thin flat plate presents similar
values of H
1 and A
3 but much smaller absolute values of A2 ; this is why the test case of Figure 8 shows soft-type flutter for the former and hard-type flutter for the latter. In conclusion, soft- and hard-type flutters have a complicated combined dynamic and aerodynamic origin and the simplified equations proposed are clearly able to distinguish between the two. Finally, it should be noted that when a 2-dof flutter calculation is performed, either rigorous or simplified, the choice of the critical modes susceptible to couple is crucial. This selection must be Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri 0 fairly complicated and this choice is not trivial. Sometimes even a multi-mode approach is necessary to properly account for the actual mechanism of instability (Øiseth et al., 2010). ⫺2 ⫺4 H*1 3. 3.1 Uncertainty in the flutter derivative measurements Flutter derivatives are usually employed in flutter and buffeting analyses as deterministic coefficients. Nevertheless, several studies have demonstrated that these type of wind tunnel measurements are characterised by significant dispersion (Mannini, 2006; Righi, 2003). This is particularly true for free-vibration setups while the problem is expected to be reduced for the forced-vibration technique where it is easier to control a few important test conditions (e.g. amplitude of vibration or length of measurements). Nevertheless, in the second case, uncertainty is not eliminated as the measurement of small aerodynamic quantities is obtained through the difference of large forces (due to the presence of inertial forces) under wind and in still-air conditions. If computational fluid dynamics simulations are employed to determine the flutter derivatives, random uncertainties are substituted by the concern of systematic errors due to the accuracy of the governing equations (in particular the turbulence model equations) or to the numerical methods used to solve them. In addition, if advanced strategies of turbulence modelling are adopted (such as three-dimensional large eddy simulation or detached eddy simulation), solutions show significant stochastic components and statistical convergence is obtained only through long and computationally expensive simulations. Therefore, if short computations are performed, the results again involve random uncertainties. ⫺6 Flatplate theory ⫺8 ⫺10 R12.5 0 2 4 6 UR (a) 8 10 0 2 4 6 UR (b) 8 10 12 0 ⫺0·2 A*2 ⫺0·4 ⫺0·6 ⫺0·8 ⫺1·0 ⫺1·2 ⫺1·4 12 5 4 A*3 3 2 1 0 0 2 4 6 UR (c) 8 10 Probabilistic approach to flutter 12 Figure 9. Comparison of flutter derivatives for a thin flat plate (Fung, 1993; Theodorsen, 1934) and a 12.5:1 rectangular cylinder (R12.5 (Matsumoto, 1996)) based on the mode coupling coefficient (Bartoli and Mannini, 2005a; Dyrbye and Hansen, 1997), which accounts for imperfect mode shape similarity, on the frequency ratio and on the absolute value of the torsional frequency. In many cases the choice is obvious in practice but sometimes more than one calculation is necessary to find the smallest flutter critical wind speed. In a few cases, especially at bridge erection stages, the mode shapes are As an example of the possible scatter in wind tunnel results,
2
Figure 10 shows the functions KH 1
, K 2 H
3 , KA2 and K A3 measured in the CRIACIV wind tunnel for a bridge deck section model (Figure 6) characterised by a geometry similar to the cross-section of Sunshine Skyway Bridge, Florida. The tests were performed according to the free-vibration technique (Mannini, 2006). The modified unifying least squares (Bartoli et al., 2009) method was used to identify the flutter derivatives and it was shown to perform very well (Bartoli et al., 2009; Mannini, 2006).
The originally measured coefficients KH 1
, K 2 H
3 , KA2 and

2


K A3 are plotted instead of H 1 , H 3 , A2 and A3 in order not to alter the experimental scatter by dividing by the reduced frequency K or K2 : It is evident that the dispersion of the data tends to increase with the reduced wind speed UR ¼ 2ð/K and becomes definitely important near the flutter boundary. This is probably due to the fact that the total damping (structural plus aerodynamic) becomes very small near flutter and therefore any disturbance has a large effect on the mechanical system. In addition, as is typical in free-vibration setups, the mean angle of attack tends to increase with increasing wind speed due to the static aerodynamic moment and is no longer negligible near flutter (Mannini, 2006; Mannini and Bartoli, 2008a). This effect implies a change in the aerodynamic properties of the crosssection and enhanced vortex shedding may be partly responsible 151 Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri 0 ⫺2·5 ⫺1 ⫺3·0 ⫺2 ⫺3·5 K 2H*3 KH*1 ⫺3 ⫺4 ⫺5 ⫺4·0 ⫺4·5 ⫺6 ⫺5·0 ⫺7 ⫺8 5 0 ⫺5·5 15 10 UR (a) 1·4 ⫺0·1 1·2 ⫺0·2 1·0 KA*2 K 2A*3 0 ⫺0·3 0·8 ⫺0·4 0·6 ⫺0·5 0 2 4 UR (c) 6 8 10 0·4 0 0 2 2 4 4 UR (b) UR (d) 6 8 10 6 8 10 Figure 10. Wind tunnel measurements of the flutter derivatives H
1 , H3
, A
2 and A
3 for the common bridge deck geometry of Figure 6 (Mannini, 2006; Mannini et al., 2010b) for the increased dispersion of the results. However, the reasons for this large scatter are not yet fully understood and, for most derivatives, this is not ascribable to the identification procedure; for instance, it could be easy to show that different values of A
2 actually correspond to signals with appreciable difference in damping in the pitching mode. The undesirable, but to a certain extent unavoidable, rolling motion of the section model (rotation about a horizontal axis parallel to the flow direction) surely represents a disturbance in the tests. Nevertheless, from analysis of the experimental data this third degree of freedom does not seem to be responsible for such a large scatter of the measurements. Other possible causes are wind tunnel free-stream turbulence, interference with the vortex-shedding excitation and the partial inadequacy of the mathematical model to explain the physical phenomenon. In particular, the previously mentioned possible non-linear dependence of the self-excited forces on the amplitude of vibration could be a source of dispersion, but only to a limited extent as a big effort was made to keep the initial condition constant during the tests and a system of electromagnets 152 was employed for this purpose (Mannini, 2006). It is worth noting that the least uncertain coefficients are H 3
and A
3 , which are related to the aerodynamic stiffness in the pitching mode. In order to characterise the flutter derivatives from a statistical point of view, relatively large samples of measurements (N ¼ 30) were obtained at three different wind speeds: a low wind speed (U ¼ 4.0 m/s); a fairly high wind speed far from flutter (U ¼ 15.0 m/s); a wind speed immediately before the instability onset (U ¼ 19.2 m/s). The first interesting point is whether the flutter derivatives are normally distributed or not. The Gaussian approximation is acceptable in most cases, although some distributions seem to be affected by a certain skewness. In particular, the normal hypothesis does not seem completely
adequate for the coefficients H
3 and A3 at low wind speed. Figure 11 shows some examples of normal probability plots for the aerodynamic coefficients A
2 and A
3 at low wind speed. Another important issue is the correlation between different Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri 0·99 0·98
no significant correlation was observed between H
1 and H 4 and


between H 2 and H 3 for the lift force, and between A1 and A
4
and between A
2 and A3 for the moment, which are proportional to the in-phase and in-quadrature parts of the forces respectively due to the heaving and pitching motion. Correlation was found


neither between H
1 and H 3 nor A1 and A3 , despite the well


known interrelations H 1 ¼ KH 3 and A1 ¼ KA3
discussed by Matsumoto (1996), Scanlan et al. (1997) and Bartoli and Mannini (2008). Finally, it seems that lift and moment flutter derivatives are mostly uncorrelated. Probability 0·95 0·90 0·75 0·50 0·25 0·10 0·05 0·02 0·01 ⫺0·054 ⫺0·053 ⫺0·052 A*2 (a) ⫺0·051 0·99 0·98 0·95 0·90 Probability 0·75 0·50 0·25 0·10 0·05 0·02 0·01 0·071 0·072 0·073 0·074 0·075 A*3 (b) Figure 11. Cumulative probability distribution for flutter derivatives A
2 and A3
at low wind speed (U ¼ 4.0 m/s) for the bridge section of Figure 6. Comparison with the corresponding normal distributions flutter derivatives. For this purpose, the correlation coefficient R was calculated for all possible couples of aerodynamic coefficients. The probability p of getting a correlation as large as the observed value by chance, when the true correlation is zero, was also determined. At low wind speed all the random variables associated with the aerodynamic derivatives seem to be uncorrelated. At U ¼ 15.0 m/s, appreciable correlation can be observed
between H 1
and H
2 (R ¼ 0.647, p ¼ 0.0001), H
3 and A3

. . (R ¼ 0 389, p ¼ 0 0336) and, above all, A1 and A2 (R ¼ 0.802, p ¼ 9.7 3 108 ). Near flutter (U ¼ 19.2 m/s), there is a strong correlation between H
1 and H
2 (R ¼ 0.967, p ¼ 3.1 3 1018 ), H
3 and H
4 (R ¼ 0.939, p ¼ 1.8 3 1014 ), A
1 and A
2 (R ¼ 0.968, p ¼ 2.1 3 1018 ) and, to a lesser extent, between A
3 and A
4 (R ¼ 0.751, p ¼ 1.7 3 106 ) and A
2 and A
4 (R ¼ 0.425, p ¼ 0.0193). Visual examples of correlation between flutter derivatives are shown in Figure 12. Interestingly, 3.2 Probabilistic model The main problem from an engineering point of view is to understand how the uncertainty in the input of a flutter calculation is transferred into the output (i.e. the critical wind speed). This is the classical problem of error propagation (see for instance the clear overview given by Näther (2009)). A deterministic calculation, which employs mean values of the flutter derivatives (or in some cases only one measured value per reduced wind speed) and structural dynamic parameters, risks becoming meaningless if a perturbation of the input is strongly amplified by the non-linear flutter equations or if the flutter critical wind speed presents a bimodal distribution (Bartoli and Mannini, 2005c). In other words, it is extremely important to estimate the variance of the critical wind speed and not just a representative deterministic value. Even better is to obtain the probability distribution of the output, which allows one to determine quantiles and confidence intervals. In addition, if X is the vector of input random variables and f is the function that relates the input and the output, the relation 23: E[ f (X )] ¼ f (E[X ]) where E[ . ] denotes the expectation operator, is in general not valid unless f is linear and it can lead to large errors for strongly non-linear problems (such as the flutter equations). As discussed in the previous section, the flutter critical wind speed depends on several dynamic and aerodynamic quantities. Nevertheless, mass, mass moment of inertia, natural frequencies and deck dimensions are known with good accuracy and Cheng et al. (2005) showed that the flutter reliability index is only slightly sensitive to the uncertainty in these random variables. By contrast, the uncertainty in the structural damping is quite high, but this parameter has a non-negligible influence on the flutter critical wind speed only in the particular case of soft-type flutter. Consequently, in this work only the flutter derivatives are considered as random variables while all the other parameters are treated as deterministic quantities. Cheng et al. (2005) showed that the flutter reliability index is very sensitive to uncertainty in the flutter derivatives. Bartoli and Mannini (2005c) considered the values of the flutter derivatives for two rectangular cylinders measured in smooth and 153 Structures and Buildings Volume 165 Issue SB3 1·2 New developments in bridge flutter analysis Mannini, Bartoli and Borri ⫺0·4 R ⫽ ⫺0·967, p ⫽ 3·1 ⫻ 10⫺18 1·0 ⫺0·5 0·8 ⫺0·6 A*2 H*2 0·6 0·4 ⫺0·7 0·2 ⫺0·8 0 ⫺0·2 ⫺13 R ⫽ ⫺0·968, p ⫽ 2·1 ⫻ 10⫺18 ⫺12 ⫺11 ⫺10 ⫺9 ⫺8 ⫺0·9 1·5 ⫺7 2·0 2·5 H*1 (a) 2·45 3·0 A*1 (b) 3·5 4·0 4·5 ⫺9 ⫺8 ⫺7 ⫺0·4 R ⫽ ⫺0·023, p ⫽ 0·902 ⫺0·5 2·40 A*3 A*2 ⫺0·6 ⫺0·7 2·35 ⫺0·8 R ⫽ ⫺0·328, p ⫽ 0·076 2·30 ⫺0·9 ⫺0·8 ⫺0·7 ⫺0·6 ⫺0·5 ⫺0·4 ⫺0·9 ⫺13 ⫺12 ⫺11 A*2 ⫺10 H*1 (c) (d) Figure 12. Examples of correlation between flutter derivatives at high wind speed (U ¼ 19.2 m/s) for the bridge section of Figure 6. R denotes the correlation coefficient, while p is the probability of obtaining a correlation as large as the observed value by chance when the true correlation is zero turbulent flow (Righi, 2003) as independent normal random variables. Monte Carlo realisations were generated in correspondence of the reduced wind speeds where measurements were available. Sets of flutter derivative functions were then obtained by interpolation. A deterministic flutter calculation was performed for each set of functions. By contrast, Mannini (2006) and Mannini and Bartoli (2009) tried to maximise the statistical perturbation of aeroelastic functions by first interpolating the mean and the variance of the measured coefficients and then generating Monte Carlo realisations of the flutter derivatives. Flutter calculations were then performed for the previously mentioned rectangular cylinders and for the realistic bridge section discussed hereafter. In this work a more rigorous probabilistic model for flutter is proposed, based on the assumption that the mechanical model of Equations 1–4 applies and the flutter derivatives are independent, 154 normally distributed random variables. As previously discussed, experimental evidence (confirmed by the results obtained for two rectangular cylinders in smooth and turbulent flow (Righi, 2003)) suggests that the flutter derivatives can be considered as normally distributed. More debatable is their statistical independency but since the correlation seems to be much lower than expected it is reasonable to waive this complication in this first application of the model. The aim is to determine the probability that a given value of the reduced wind speed UR ¼ 2ð=K is larger than the critical reduced wind speed, p(UR > URc ); that is, flutter occurs at that reduced wind speed. From a theoretical point of view, it could be possible to solve the probabilistic flutter problem by considering the flutter equations and calculating the probability that the solution of the so-called imaginary equation (real values of Y ¼ ø=øh , such that 1 < Y < ªø , for which the imaginary part of the determinant of Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri the system of equations obtained imposing coupled harmonic oscillations vanishes) is larger than (or equal to) that of the real equation (values of Y for which the real part of the determinant vanishes); this corresponds to an unstable condition (Mannini, 2006) if the common form of fourth- and third-order equations (as reported for instance in Dyrbye and Hansen (1997)) is used (see Figure 13). Otherwise, if the form of Equations 10 and 11 is employed, instability is reached when the solution of the imaginary equation (Equation 11) is smaller than that of the real equation (Equation 10). This probability could be calculated through the formulas for probability density functions (PDFs) of the product, the sum and the square root of continuous random variables. Nevertheless, in the general case of 2-dof flutter, this analytical procedure becomes too complicated and burdensome. It is applicable only in the case of 1-dof torsional flutter or if an approximate model such as that of Equation 17 or Equation 21 is considered. Based on Equation 21, the probability of the flutter condition would read: speed is characterised by the same probability function as the reduced critical wind speed, the probability distribution and the statistical moments of the former can be simply calculated through the relation U ¼ Bøc UR =2ð, where the critical frequency øc is deterministically calculated. 24:  p(UR > URc ) ¼ p A
2 (U R ) > 2Ig Æ rB4  25: Much more practical for engineering applications is the use of Monte Carlo strategies. At each reduced wind speed a set of flutter derivatives is generated from the known PDFs. For every realisation, the flutter condition is verified by comparing the solutions of the real and imaginary equations and the probability of encountering flutter is obtained. By spanning all the reduced wind speeds in the range of interest, the cumulative probability distribution function (CDF) p(URc < UR ) can be determined. Furthermore, if one assumes that the dimensional critical wind 1·9 1·8 1·7 Y Finally, it is worth noting that the CDF of the flutter critical wind speed p(Uc < U ) expresses the probability of encountering flutter given a certain wind speed. Therefore, such a function is suitable to be used in the framework of the performance-based design, accounting for the bridge collapse limit state due to flutter. In fact, considering the uncertainty in hazard, vulnerability and damage, a bridge structure can be conceived according to the Pacific Earthquake Research Centre (PEER) equation (Ciampoli et al., 2009; Porter, 2003). In particular, once a limit state related to the structural response has been assumed as a decision variable, this equation simplifies to (Mannini, 2006): Pfail  p(EDP) ¼ ð p(EDP j IM)  g(IM) d(IM)
where Pfail is the probability of failure, EDP is an engineering demand parameter representing the structural response with which the considered limit state is quantified, IM is the intensity measure of the hazard, p( . ) denotes the probability of exceedance, p( . | . ) is a conditional probability of exceedance and g(IM) is the probability density function of the hazard. The integral is evaluated over the entire space
of possible intensity measures. Considering the ultimate limit state for flutter, assuming a probability of 1 for the collapse of the bridge structure under flutter vibrations and taking the mean wind speed as the intensity measure, the structural vulnerability term p(EDP|IM) becomes the probability of encountering flutter given a certain wind speed. Therefore p(EDP|IM) is the CDF of the flutter critical wind speed. As a consequence, determining the probability function of the flutter critical wind speed is important not only in order to estimate the variance of the results of a flutter calculation but also to account for this aeroelastic instability phenomenon in a risk analysis. 1·6 1·5 Real equation Imaginary equation 1·4 1·3 0 2 4 6 8 10 UR Figure 13. Deterministic solution of the real and imaginary flutter equations. Y ¼ ø=ø h is the non-dimensional flutter frequency and ªø ¼ øÆ =ø h is the still-air frequency ratio. Real and imaginary equations are those reported in several texts (e.g. Dyrbye and Hansen, 1997). ªø ¼ 1:948 3.3 Example of application As an example of application of the outlined probabilistic flutter approach, the wind tunnel section model depicted in Figure 6 is taken as a case study (Mannini, 2006). In order to give a statistical description of the flutter derivatives, measurements were repeated ten times for each reduced wind speed while trying to keep the test conditions unchanged. Four out of eight aerodynamic coefficients are shown in Figure 10, even though the complete set of eight flutter derivatives was used in both deterministic and probabilistic calculations. Mean and standard deviation values of the flutter derivatives are interpolated and a very limited extrapolation up to UR ¼ 10 was necessary. The basic parameters of the mechanical system are: B ¼ 0.45 m, m ¼ 5.45 kg/m, I ¼ 0.095 kg m2 /m, ø h ¼ 19.20 rad/s, øÆ ¼ 37.40 rad/s, æ h ¼ 0.24% and æÆ ¼ 0.18%. 155 Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri 1·0 The graphical deterministic solution of the flutter equations, obtained by considering the mean values of the flutter derivatives, is shown in Figure 13. Table 7 shows that the calculated critical value agrees very well with the experimentally measured value. Calculated Normal 0·8 0·7 0·6 CDF The CDF of the critical wind speed was obtained through Monte Carlo simulation with 500 000 realisations for each reduced wind speed. The reasonable convergence of the results with respect to the number of realisations was verified. The range of reduced wind speed spanned was from 0 to 10 in increments of 0.02. The PDF was obtained from the CDF by central differencing. Both functions are reported in Figure 14. Comparison with the corresponding normal distributions shows that the result is not perfectly Gaussian, with a small positive skewness and a kurtosis smaller than 3.0. Moreover, it is possible to remark that the probability of encountering flutter is zero for reduced wind speeds lower than 8.5, whilst it is 1.0 for reduced wind speeds larger than 10. 0·9 0·5 0·4 0·3 0·2 0·1 0 8·0 8·5 9·0 9·5 10·0 9·5 10·0 UR (a) 1·8 1·6 It is also worth noting that, due to the non-linearity of the flutter equations, a large dispersion characterising several flutter derivatives is transformed in this case into a small dispersion of the critical wind speed. In fact, in the range of interest of reduced wind speed, coefficients of variation up to about 8% were observed for H
1 and A
1 , up to 3–4% for H
3 and A
3 , up to 30% for A
2 and even beyond 40, 50 and 100% respectively for

H
2 , A4 and H 4 : For the same reason, despite the normal 1·4 1·2 PDF Table 7 lists the first four statistical central moments. It is evident that the mean value of the reduced wind speed is in this case very close to the one calculated in a deterministic way and to the wind tunnel result. Nevertheless, the probabilistic approach also reveals that the coefficient of variation is small (less than 3%); that is, large deviations of the solution from the mean are not likely, similarly to the previous predictions of error propagation (Bartoli and Mannini, 2005c; Mannini and Bartoli, 2009). In these conditions, the mean value of the distribution or the result of a deterministic calculation is a reliable estimate of the flutter critical wind speed, although the 95% confidence interval for the reduced critical wind speed is (8.86, 9.86), the amplitude of which is not negligible for engineering purposes. 1·0 0·8 0·6 0·4 0·2 0 8·0 8·5 9·0 UR (b) Figure 14. Cumulative probability distribution function (CDF) and probability density function (PDF) of the reduced critical wind speed for the case study considered. The corresponding normal distributions are also reported for comparison distribution of the input, the output of the calculation is not perfectly Gaussian. Experimental 9.31 Deterministic 9.42 Probabilistic ì CoV: % 9.36 2.69 skw krt 0.22 2.64 Table 7. Comparison between the reduced critical wind speed measured in the wind tunnel for the test case considered, the result of the deterministic flutter calculation and those of the probabilistic approach in terms of mean value ( ì), coefficient of variation (CoV), skewness (skw) and kurtosis (krt) 156 Finally, it must be noted that the well-behaving results (mean value of the critical wind speed close to the deterministic value, small variance, mono-modal distribution, etc.) observed for the particular case study considered can in no way be extended to different test cases. 4. Conclusion This paper has shown that the problem of flutter stability can be analytically simplified by manipulating the flutter equations on the basis of experimental evidence. The procedure leads to two approximate equations through which the critical reduced wind Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri speed and the flutter frequency can be calculated with just three flutter derivatives instead of the usual eight coefficients. These formulas give accurate results provided that the frequency separation of the critical modes is not too small. They represent a considerable simplification and allow a deeper understanding of the flutter mechanism and consequent better tailoring of a bridge structure at early design stages. In particular, the proposed formulas help to explain the role played by damping and other structural parameters in the onset of instability and to give an explanation for soft- and hard-type flutter. Document. See http://www.aniv-iawe.org/barc for further details (accessed 27/08/2011). Bartoli G, Contri S, Mannini C and Righi M (2009) Towards an improvement in the identification of bridge deck flutter derivatives. ASCE Journal of Engineering Mechanics 135(8): 771–785. Bisplinghoff RL, Ashley H and Halfman RL (1996) Aeroelasticity, 1st edn. Dover Publications, New York. Caracoglia L and Jones NP (2003) Time domain vs. frequency domain characterisation of aeroelastic forces for bridge deck sections. Journal of Wind Engineering & Industrial Aerodynamics 91(3): 371–402. Chen X (2007) Improved understanding of bimodal coupled bridge flutter based on closed-form solutions. ASCE Journal of Structural Engineering 133(1): 22–31. Chen X and Kareem A (2003) Efficacy of tuned mass dampers for bridge flutter control. ASCE Journal of Structural Engineering 129(10): 1291–1300. Chen X and Kareem A (2006) Revisiting multimode coupled bridge flutter: some new insights. ASCE Journal of Engineering Mechanics 132(10): 1115–1123. Chen X, Matsumoto M and Kareem A (2000) Aerodynamic coupling effects on flutter and buffeting of bridges. ASCE Journal of Engineering Mechanics 126(1): 17–26. Cheng J, Cai CS, Xiao RC and Chen SR (2005) Flutter reliability analysis of suspension bridges. Journal of Wind Engineering & Industrial Aerodynamics 93(10): 757–775. Chowdhuri AG and Sarkar PP (2004) Identification of eighteen flutter derivatives of an airfoil and a bridge deck. Wind and Structures 7(3): 187–202. Ciampoli M, Petrini F and Augusti G (2009) A procedure for performance-based wind engineering. Proceedings of 10th International Conference on Structural Safety and Reliability, Osaka. Taylor & Francis, London, pp. 1843–1850. Costa C and Borri C (2006) Application of indicial functions in bridge deck aeroelasticity. Journal of Wind Engineering & Industrial Aerodynamics 94(11): 859–881. Diana G, Resta F, Zasso A, Belloli M and Rocchi D (2004) Forced motion and free motion aeroelastic tests on a new concept dynamometric section model of the Messina suspension bridge. Journal of Wind Engineering & Industrial Aerodynamics 92(6): 441–462. Diana G, Resta F and Rocchi D (2008) A new numerical approach to reproduce bridge aerodynamic non-linearities in time domain. Journal of Wind Engineering & Industrial Aerodynamics 96(10–11): 1871–1884. Dyrbye C and Hansen SO (1997) Wind Loads on Structures, 1st edn. Wiley, New York. Frandsen AG (1966) Wind stability of suspension bridges. Application of the theory of ‘thin airfoils’. Proceedings of International Symposium on Suspension Bridges, Lisbon. National Laboratory for Civil Engineering, Lisbon, pp. 609– 627. Fung YC (1993) An Introduction to the Theory of Aeroelasticity. Dover Publications, New York. The problem of the uncertainty observed in wind tunnel measurements of flutter derivatives (especially in the case of freevibration setups) was discussed. A model to account for the nondeterministic nature of these coefficients in the calculation of the flutter critical wind speed was presented. The main purpose of this model is estimation of the variance of the calculated critical wind speed, which is absolutely unknown by performing the usual deterministic flutter analysis. An application in the case of a bridge deck section model of common geometry was outlined, employing wind tunnel data measured ad hoc for this purpose. As a future development of this work it will be interesting to consider the actual correlation observed between a few flutter derivatives. Finally, it is worth remarking that the approximate formulas for the critical wind speed can also be fruitfully employed in order to solve analytically the problem of probabilistic flutter assessment. Acknowledgements This research work was partially performed in the framework of the Italian National Research Contracts AER-BRIDGE (PRIN 2006) and Wi-POD (PRIN 2007), 2-year grants from the Italian Ministry of Education, University and Research (MIUR). REFERENCES Bartoli G and Mannini C (2005a) From multimodal to bimodal approach to flutter. Proceedings of 6th European Conference on Structural Dynamics, Paris. Millpress, Rotterdam, pp. 349–354. Bartoli G and Mannini C (2005b) A simplified method to evaluate the critical flutter speed for flexible bridge decks. Proceedings of 4th European and African Conference on Wind Engineering, Prague. ITAM ASCR, Prague. Bartoli G and Mannini C (2005c) Reliability of bridge deck flutter derivative measurements in wind tunnel tests. Proceedings of 9th International Conference on Structural Safety and Reliability, Rome. Millpress, Rotterdam, pp. 1193–1200. Bartoli G and Mannini C (2008) A simplified approach to bridge deck flutter. Journal of Wind Engineering & Industrial Aerodynamics 96(2): 229–256. Bartoli G and Righi M (2006) Flutter mechanism for rectangular prisms in smooth and turbulent flow. Journal of Wind Engineering & Industrial Aerodynamics 94(5): 275–291. Bartoli G, Bruno L, Buresti G et al. (2008) BARC Overview 157 Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri Ge YJ, Xiang HF and Tanaka H (2000) Application of a reliability The complex branch characteristics of coupled flutter. Journal of Wind Engineering & Industrial Aerodynamics 96(10–11): 1843–1855. Nakamura Y (1978) An analysis of binary flutter of bridge deck sections. Journal of Sound and Vibration 57(4): 471–482. Nakamura Y (1979) On the aerodynamic mechanism of torsional flutter of bluff structures. Journal of Sound and Vibration 67(2): 163–177. Nakamura Y and Mizota T (1975) Torsional flutter of rectangular prisms. Journal of the Engineering Mechanics Division ASCE 101(2): 125–142. Näther W (2009) Copulas and T-norms: Mathematical tools for modeling propagation of errors and interactions. Proceedings of 10th International Conference on Structural Safety and Reliability, Osaka. Taylor & Francis, London, pp. 1238– 1245. analysis model to bridge flutter under extreme winds. Journal of Wind Engineering & Industrial Aerodynamics 86(2–3): 155–167. Jain A, Jones NP and Scanlan RH (1996) Coupled flutter and buffeting analysis of long-span bridges. ASCE Journal of Structural Engineering 122(7): 716–725. Katsuchi H, Jones NP and Scanlan RH (1999) Multimode coupled flutter and buffeting analysis of the Akashi Kaikyo Bridge. ASCE Journal of Structural Engineering 125(1): 60–70. Lin YK (1996) Stochastic stability of wind-excited long-span bridges. Probabilistic Engineering Mechanics 11(4): 257– 261. Mannini C (2006) Flutter Vulnerability Assessment of Flexible Bridges. PhD thesis, University of Florence–TU Braunschweig. Mannini C and Bartoli G (2008a) Investigation on the dependence of bridge deck flutter derivatives on steady angle of attack. Proceedings of 5th International Colloquium on Bluff Bodies Aerodynamics and Applications, Milan. Politecnico di Milano, Milan. Mannini C and Bartoli G (2008b) Analisi del meccanismo del flutter per impalcati da ponte: sensibilità rispetto ai parametri dinamici della struttura. Proceedings of 10th Italian National Conference on Wind Engineering, Cefalù, Italian National Association of Wind Engineering, Rome (in Italian). Mannini C and Bartoli G (2009) The problem of uncertainty in the measurement of aerodynamic derivatives. Proceedings of 10th International Conference on Structural Safety and Reliability, Osaka. Taylor & Francis, London, pp. 824–831. Mannini C, Bartoli G, Flamand O and Grillaud G (2007) A simplified method of flutter assessment: first steps towards flutter derivative generalization. Proceedings of 12th International Conference on Wind Engineering, Cairns, Australasian Wind Engineering Society, pp. 2343–2350. Mannini C, Ferrucci M and Bartoli G (2010a) Effetti non lineari nelle derivate di flutter di un impalcato da ponte. Proceedings of 11th Italian National Conference on Wind Engineering, Spoleto, Morlacci, Perugia (in Italian). Mannini C, Šoda A, Voâ R and Schewe G (2010b) Unsteady RANS simulation of flow around a bridge section. Journal of Wind Engineering & Industrial Aerodynamics 98(12): 742– 753. Matsumoto M (1996) Aerodynamic damping of prisms. Journal of Wind Engineering & Industrial Aerodynamics 59(2–3): 159–175. Matsumoto M, Yoshizumi F, Yabutani T, Abe K and Nakajima N (1999) Flutter stabilization and heaving-branch flutter. Journal of Wind Engineering & Industrial Aerodynamics 83(1–3): 289–299. Matsumoto M, Taniwaki Y and Shijo R (2002) Frequency characteristics in various flutter instabilities of bridge girders. Journal of Wind Engineering & Industrial Aerodynamics 90(12–15): 1973–1980. Matsumoto M, Okubo K, Ito Y, Matsumiya H and Kim G (2008) 158 Noda M, Utsunomiya H, Nagao F, Kanda M and Shiraishi N (2003) Effects of oscillation amplitude on aerodynamics derivatives. Journal of Wind Engineering & Industrial Aerodynamics 91(1–2): 101–111. Øiseth O, Röonquist A and Sigbjörnsson R (2010) Simplified prediction of wind-induced response and stability limit of slender long-span suspension bridges, based on modified quasi-steady theory: a case study. Journal of Wind Engineering & Industrial Aerodynamics 98(12): 730– 741. Ostenfeld-Rosenthal P, Madsen HO and Larsen A (1992) Probabilistic flutter criteria for long span bridges. Journal of Wind Engineering & Industrial Aerodynamics 42(1–3): 1265–1276. Porter KA (2003) An overview of PEER’s performance-based engineering methodology. Proceedings of 9th International Conference on Applications of Statistics and Probability in Civil Engineering, San Francisco. IOS Press, Amsterdam, pp. 973–980. Pourzeynali S and Datta TK (2002) Reliability analysis of suspension bridges against flutter. Journal of Sound and Vibration 254(1): 143–162. Righi M (2003) Aeroelastic Stability of Long Span Suspended Bridges: Flutter Mechanism on Rectangular Cylinders in Smooth and Turbulent Flow. PhD thesis, University of Florence. Scanlan RH (1978a) The action of flexible bridges under wind, I: Flutter theory. Journal of Sound and Vibration 60(2): 187– 199. Scanlan RH (1978b) The action of flexible bridges under wind, II: Buffeting theory. Journal of Sound and Vibration 60(2): 201– 211. Scanlan RH and Tomko JJ (1971) Airfoil and bridge deck flutter derivatives. Journal of the Engineering Mechanics Division ASCE 97(6): 1717–1737. Scanlan RH, Jones NP and Singh L (1997) Inter-relations among flutter derivatives. Journal of Wind Engineering & Industrial Aerodynamics 69–71: 829–837. Selberg A (1961) Oscillation and aerodynamic stability of Structures and Buildings Volume 165 Issue SB3 New developments in bridge flutter analysis Mannini, Bartoli and Borri suspension bridges. Acta Polytechnica Scandinavica: Civil Engineering and Building Construction Series, 13(308): 1–69. Simiu E and Scanlan RH (1996) Wind Effects on Structures: Fundamentals and Applications to Design, 3rd edn. Wiley, New York. Singh L, Jones NP, Scanlan RH and Lorendeaux O (1995) Simultaneous identification of 3-dof aeroelastic parameters. Proceedings of 9th International Conference on Wind Engineering, New Delhi. Wiley Eastern, New Delhi, pp. 972–981. Theodorsen T (1934) General Theory of Aerodynamic Instability and the Mechanism of Flutter. NACA, Langley Research Center, Hampton, Technical report 496, Annual report 20. Vairo G (2010) A simple analytical approach to the aeroelastic stability problem of long-span cable-stayed bridges. International Journal for Computational Methods in Engineering Science & Mechanics 11(1): 1–19. WHAT DO YOU THINK? To discuss this paper, please email up to 500 words to the editor at [email protected]. Your contribution will be forwarded to the author(s) for a reply and, if considered appropriate by the editorial panel, will be published as a discussion in a future issue of the journal. Proceedings journals rely entirely on contributions sent in by civil engineering professionals, academics and students. Papers should be 2000–5000 words long (briefing papers should be 1000–2000 words long), with adequate illustrations and references. 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