In this article we will discuss about:- 1. Introduction to Buffeting Theories on Long-Span Cable Supported Bridges 2. Refinement of Buffeting Theory 3. Buffeting Response Spectrum Method 4. Buffeting Response Analysis in Time Domain 5. Passive TMD Control of Buffeting 6. Conclusion.
Introduction to Buffeting Theories on Long-Span Cable Supported Bridges:
The wind-induced vibration of long-span cable supported bridges (cable-stayed bridge and suspension bridge) behaves mainly as the flutter and buffeting response of deck as well as the wind-induced vibration of cables. Flutter is a self-excited divergent vibration, but a well-designed cross section of deck may have large enough critical wind speed in order to avoid the occurrence of such catastrophic failure. The local wind-induced vibration of cables can also be suppressed through some effective measures for vibration reduction.
With the increase of span length, the bridge structure tends to be more flexible. The excessive buffeting response under the action of near-ground turbulent wind, although it is not destructive, may cause fatigue problems due to high frequency of occurrence and traffic discomfort, even lead to the un-safety of passing vehicles with high speed.
The method developed by Davenport and Scanlan are generally adopted in the buffeting response analysis. Davenport introduced the aerodynamic admittance function and joint acceptance function in order to modify the buffeting force spectrum, but considered simply the self-excited forces, Oppositely, Scanlan considered completely the coaction of self-excited forces. These two theories have the same assumption that the buffeting response is linear, and the superposition of modal responses can be used.
Refinement of Buffeting Theory:
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The buffeting response of actual bluff bridge deck under turbulent flow is a complicated problem, because both the approach flow turbulence and the body signature turbulence interacted with the motion of structure will influence the buffeting force to a greater or lesser extent.
The conventional frequency-domain buffeting analysis framework is based on the quasi-steady assumption with a consideration of aerodynamic admittance for the non-steady effect.
Although the frequency-domain approach is successfully applied to the buffeting analysis of many long-span bridges, the refinement of buffeting theory for more accurate response prediction is necessary in following aspects:
(1) Flutter derivatives, especially the aerodynamic damping terms, strongly influence the buffeting response. So the effect of turbulence on flutter derivatives should be further studied.
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(2) Aerodynamic admittance functions tor commonly used bridge deck should be identified and classified into several categories and, if possible, given approximately by formulae.
(3) Correlation index λ in wind speed spectrum is an important value for the accuracy of buffeting response prediction. Field measurement of space correlation of wind speed on the existed long-span bridges should be done.
(4) Interaction between turbulence components and motion components of bridge deck should be investigated through theoretical analysis for assessing their effects on buffeting response.
(5) Coupling effect between motions and vibration modes should be analyzed especially at the erection stage.
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(6) Finally, the time-domain analysis would be a useful approach in considering both aerodynamic and structural non-linear effects.
Buffeting Response Spectrum Method:
The theoretical analysis and field measurement show that the influence of aerodynamic coupling between modes is negligible. Therefore, for the case of vertical bending, lateral bending and mainly torsional mode, we can take each mode for estimating separately their buffeting response. This situation is similar to the seismic analysis by using the response spectrum method.
In order to make the estimation of buffeting response of long-span bridges simple and clear, and to assess easily the aerodynamic behaviour of cross section from the viewpoint of buffeting response in the preliminary design stage, it is possible to establish a practical calculation method for the buffeting response spectrum.
Fundamental Formulae for Buffeting Response Spectrum:
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Based on the random vibration theory, and by ignoring the aerodynamic coupling between the different motion components, and the cross spectra between horizontal fluctuating wind u(x, t) and vertical one w(x, t), the RMS response spectrum of all three motions can be written as follows:
In which, the torsional response spectrum has been multiplied by B/2 (half width of bridge deck) for the unity of dimension in order to make comparison with vertical displacement h and lateral one p; SL(ω), SM(ω) and SD(ω) are the buffeting force spectra for three direction considering the aerodynamic admittance function respectively; 
are the joint acceptance functions corresponding to three generalized coordinates for the basic mode shapes h(t), α(t) and p(t) respectively, which reflect the space correlation of fluctuating wind, 
are the aerodynamic transfer functions for three displacement degrees of freedom, where the effects of self-excited forces (aerodynamic derivatives) are included in frequency ω̅ and damping ratio z̅.
If only the resonance part is taken into account, the calculation formulae for buffeting response spectrum expressed by Eqs.(1) can be simplified as follows:
Practical Calculation Formulae for Buffeting Response Spectrum:
In the conventional seismic analysis of bridges, the response spectrum method is often used. In order to make the estimation of buffeting response simple and clear such as the calculation of seismic response, it is necessary to neglect some secondary factors, so as to simplify further the fundamental formulae. The buffeting response spectrum of vertical bending mode is taken as an example for explanation.
(1) The dimensionless reduced frequency k = (ω. B)/U is introduced as a self- variable, and then the RMS buffeting response spectrum for h(t) can be rewritten as follows:
(3) The joint acceptance function |Jh (kh)|2 is a dual integration.
If the first mode shape of deck can be described approximately by an equivalent sine function, the expression can be obtained as follows:
According to the analysis and simplification mentioned above, the practical calculation formula for RMS buffeting response spectrum of vertical bending modes can be obtained as follows:
Related to the dimension of structure, inertia, damping ratio and the aerodynamic derivative as well, which reflects the total damping level of structure; Ψ(β) is a coefficient related to the joint acceptance function. It can be calculated by equivalent length of sine function L, see Fig. 1; µ (kh) is a coefficient related to the aerodynamic admittance function, see Fig.2, фw(f) = 1/(1 + 4f), (1 = n . z/U) is a coefficient related to the vertical wind spectrum, see Fig.3.
Similarly, the practical calculation formulae from RMS buffeting response spectrum of torsion and lateral bending can be written as:
Buffeting-based Selection of Deck Section:
Aerodynamic selection of deck cross section shape is very important in the preliminary design stage of a long-span bridge. In the past years, this selection mainly aimed at flutter-based selection, but it is better to further choose one section according to the buffeting characteristics in several cross sections which are all satisfactory from flutter-based selection.
It can be assumed that the difference of deck cross section shapes have only effects on the aerodynamic loads acting on each deck, while have no effects on the structural dynamic behaviour, such as Eigen frequencies, mode shapes, etc.
As a comparative study, a basic reference cross section is first chosen from the provided cross sections in buffeting-based selection, and others are compared with it.
Let the RMS buffeting responses of the reference one to be (Rh1)b, (Rp1)b and (Ra1)b, and the others to be (Rb1)1, (Rp1), and (Ra1)i, “comparable buffeting coefficient” are defined and derived as follows:
Where Z is the bridge deck level.
For further simplification of the method, ƞsh and ƞsa can be expressed as:
As the condition of ƞuw = 2~4, CL‘> > CL, CM‘ > > CM and CL‘> > (A/B)CD for many bridges. It means that, for a relative comparison, the slope of the aerodynamic force coefficient curve of the deck section CL‘ and CM‘ are the predominant parameters in the buffeting-based selection.
Buffeting Response Analysis in Time Domain:
In order to investigate the response characteristics of a long span bridge, such as the effects of instantaneous relative velocity and the effective angle of attack, and structural nonlinearity, some researchers have utilized the time domain approach. J. C. Santos (1993) and I. Kovacs (1992) have presented solution method in their papers Wind forces are modeled by using the quasi-steady formulation which takes instantaneous effective angle of attack and time-space variation of wind velocity fluctuation into account.
In the time-domain analysis, turbulent components u(x, t) and w(x, t) and buffeting forces are transformed into time history using computer simulation, in which time- space variation of wind velocity fluctuation is considered. The self-excited forces appear to be in the form of convolution between impulse function and response, whose parameters are determined by fitting aerodynamic derivatives obtained in bridge section model wind tunnel test.
Furthermore, the instantaneous change of drag, moment and lift coefficients with the effective angle of attack, especially the wind velocity turbulence effect on aerodynamic loads are considered, which cannot be performed in the frequency domain.
Simulation of Wind Turbulent Components:
The wind velocity at one point is assumed to compose of a mean part U, and a fluctuating part u(x, t) for the along wind direction and a fluctuating part w(x, t) for the vertical direction. When performing the time domain analysis of long span bridge under wind action, the time-space variation of wind velocities along the bridge axis is concerned, i.e., turbulent components n(x, t) and w(x, t) are modeled as multi-correlated random process.
Simulation of multi-correlated random processes can be done in several ways, such as methods based on the summation of trigonometric function (wave superposition) and methods based on the convolution of white noise with a kernel function or integration of a differential equation driven by white noise (digital filtering).
These techniques vary in their applicability, complexity, computer storage requirement, and computing time. Due to its good versatility, wave superposition method based on a prescribed correlation function and power spectral density function is obtained here.
To comply with the coherence criteria, the turbulent part of the momentary wind velocity in point x is generated as the following sum:
where, 
k is decaying coefficients, S(n) is target spectrum; zk is generation spot; ωj is circular frequency, δω1 is circular frequency to avoid periodicity of generated random process, δ ω ϵ(-Δω’/2, Δω’/2), Δω ‘<< Δω; фkj is starting phase, uniformly distributed between 0 and 2π; n is frequency (Hz).
For the longitudinal component u(x, t) of the wind velocity fluctuation, the turbulence spectrum is defined, according to Simiu, as:
Where, f = n z|U(z), is reduced frequency; u. = KU(z)/In[(z – zd)/z0], K is Karman constant(= 0.4), zd = 0, in open zone; z0 is roughness length of the ground surface (0.003m), z is height; U(z) is mean velocity at height z.
On the other hand, for the vertical component of velocity fluctuation, the Panofsky-McCormick spectrum is expressed as:
Where, f = n z/U(z), the remaining symbols are the same as the Eq.(21).
For the Shantou Bay Bridge, the turbulent components u(x, t) of a mean wind speed U = 30m/s are simulated along the bridge span, shown in Fig. 5. Comparison between the target spectra and the spectra obtained from the simulated velocities is shown in Fig. 6. It can be seen that velocity fluctuation can be simulated satisfactorily using Eq.(20).
At a particular time instant, buffeting forces derived from the quasi-steady formulation are expressed as:
Where, A is cross-wind projected area (per unit span) normal to U̅; α0 is the effective angle of attack under the wind velocity; CD, CL, CM are instantaneous drag, lift and moment coefficients per unit span, respectively; CL‘, CM‘ are instantaneous slope of CL and CM at angle instant of attack; ρ is air density, B is bridge deck width.
Taking aerodynamic admittance and correlation function to Eq.(23), and making the Fourier transform, the buffeting forces spectra are obtained as following:
Assuming [SL], [SM], and [SD] are buffeting force – lift, moment and drag – spectral density matrix respectively, they can be expressed as the following:
According to R. H. Scanlan and Y. K. Lin, self-excited loads are assumed to be generated by linear mechanisms and expressed in terms of convolution integrals as follows:
Assuming that the surrounding air flow provides a set of filter-like devices in generating the buffeting forces on a bridge, the transfer function FMa may be approximated as follows for the purpose of fitting the experimental data.
Where, c1, c2, c3, c4, c5, c6 are real dimensionless coefficients to be inferred from measured flutter derivatives, U is instantaneous total wind speed (U̅ + u).
Time History Analysis of Buffeting Response:
The equations governing the motion of a bridge deck is as follows:
where, Lb(t),Mb(t),Db(t) are generalized fluctuating components-induced lift, moment and drag, respectively; h, α, p are generalized vertical bending, torsional and swaying displacements, respectively; zh, za , zp are structural damping ratios; ωh. ωa, ωp are structural circular frequencies; mh, Ia, mp are generalized mass; Lm(t)Mse(t),Dse(t) are self-excited force.
In the analysis, the wind fluctuation along the bridge axis are generated using Eq.(20) derived from the given Simiu spectrum, Panofsky-McCormick spectrum and Davenport coherence function. At each time increment, the drag, lift and moment coefficients vary with the effective angle of attack.
The Shantou Bay Bridge’s coupled buffeting response of vertical bending and torsion motions corresponding to fundamental vibration mode are calculated as a result, shown in Fig. 7 and Fig. 8 Fig. 9 demonstrates a good accordance among the results of buffeting responses calculated in time domain and frequency domain, and measured by 3D full scale wind tunnel test.
Passive TMD Control of Buffeting:
At present, the measures which are adopted for increasing the critical flutter wind speed of bridges include improving the aerodynamic behaviour mainly by modifying the cross section shape, and increasing the first torsional frequency by using inclined cable-planes. However, buffeting is still a problem during erection and operation.
Many engineers and researchers are concerned about how to suppress the buffeting of bridges effectively. TMD (tuned mass damper) devices with the advantages of reasonable cost and convenience, and effectiveness in usage, have been mounted on some bridges, TMD for increasing the flutter and galloping critical wind speeds as well as suppressing buffeting of long- span bridges have been theoretically and experimentally studied.
Control of Vertical Bending Buffeting:
For bridges with a higher first torsional frequency, the contribution from torsional modes to the total buffeting response may be neglected Therefore, it can be assumed that the total buffeting response only takes place in vertical bending modes, and further in the first vertical bending mode, so ф1(x), generally speaking, occupies a dominant position in the total response
The mechanical model for analysis is shown in Fig. 10(b). The girder displacement and the TMD displacement relative to the girder can be written as Y1(x, t) = ф1(x)z1(t) and Y2(t) = ф1(x0)z2 (t) respectively.
The equations governing the motions of a bridge deck and TMD are, respectively, as follows:
Where z1 and z2 are the first mode time-dependent vertical generalized coordinates of the bridge deck and TMD, respectively, ф1(x0) is the value of ф1(x) at the site where TMD is mounted; M1, ω1 and z1 are the first mode generalized mass, circular frequency and damping ratio of bridge, respectively, ω2 and z2 are the circular frequency and the damping ratio of TMD, respectively, µ = m2ф21(x0)/M1 is the generalized mass ratio of TMD to the bridge; m2 is the mass of TMD; Ls is the total length of the main bridge, Fae(x, t) and Fb (x, t) are the aeroelastic force and buffeting force acting on the bridge deck proposed by Scanlan.
From the Eqs. (29) and (30), the mean square displacements of the bridge deck and TMD can be easily derived respectively.
Then, is defined as the reduction ratio, where Ry1 and Ry denote the RMS displacements of the bridge deck with and without TMD respectively.
Eq.(31) is here treated as the objective function for optimization of parameters of TMD.
The equations which describe how the optimal damping ratio (z20) and frequency (f20) depend on the generalized mass ratio of TMD to bridge (µ) and z1 are given as follows:
Where f1 is the vertical bending Eigen-frequency of structure which need to be controlled, z1 is the total damping ratio, the sum of structural one and aerodynamic one, which is a function of mean wind speed U̅.
Eqs.(32) and (33) are the same as the classical equations for very small. But when is much larger than the inherent structural one, the classical equation will affect the optimal frequency of TMD to a certain extent.
By using the Least Square Method, an expression for the optimal reduction ratio corresponding to the optimal parameters from Eqs.(32) and (33) is derived as follows:
It should be noted that ƞ0 may be much larger at allowable wind speed than at a high one because it is inverse proportional to z1, which depends directly on wind speed. For these reasons, it is suggested that in designing TMD for a given bridge, designers first should select a “TMD design speed”, at which the parameters of TMD are optimized.
When engineers design a TMD for a given bridge and a desired reduction ratio, they can determine a value of µ (mass of TMD) of a selected “TMD design speed” by using Eq.(35) at first. Then, from Eqs.(32) and (33), the optimal parameters of TMD can be easily found.
The Yangpu Bridge [28] is a cable-stayed bridge with a composite deck, include cable planes and a center span of 602m. The first vertical bending frequency is 0.305Hz. The inherent structural damping ratio is taken as 0.01. According to the results of buffeting analysis for the bridge before closure, the required reduction ratio is at least 35% at “TMD design wind speed” of 25m/s The calculated total mass of TMD using Eq.(35) is about 40 tons, and the optimal frequency and damping ratio using Eqs.(32) and (33) are 0.298 and 0.072 respectively.
Control of Buffeting Response with Vertical Bending — Torsional Modes Interaction:
For some cases that the vertical buffeting and torsional one are of the same significance, the control of buffeting response should take the interaction between two modes into account Fig. 10(c) shows a mechanical model for the control of interaction buffeting, in which Y1 and α denote vertical displacement and torsional angle of the bridge deck respectively; Y2 and Y3 denote vertical displacements of TMD1 and TMD2 relative to Y1 respectively. TMD1 and TMD2 have the same mass, spring stiffness and damping ratio, that is, M2 = M3, K2 = K3 and C2 = C3 (z2= z3).
The Objective function for optimization of the TMDs’ parameters are taken as the mean square displacement (R2B) comprising vertical value (R2b) and torsional angle (R2t) times (B/2)2, where B is the width of bridge deck, that is.
The amplitudes of TMD can be taken into consideration of the constraint equations. If the bridge vibrates only in vertical motion, the vibration amplitudes of TMD1 and TMD2 are the same. But the existence of torsional motion leads to the situation that the amplitudes of TMD1 and TMD2 are not the same. Larger RMS (root mean square) displacement between TMD1 and TMD2 can be found and used to check the engineering constraint condition.
In general, the optimal frequency of TMD is near one frequency on which the buffeting energy concentrates mainly. When the frequency of TMD varies apart from the optimal value between the vertical and torsional Eigen-frequencies, the reduction ratio decreases slowly. While it is away from the optimal value outside the two frequencies, the reduction ratio falls down quickly.
Constraint Condition:
It is important to establish the engineering constraint conditions and to take them into consideration in designing TMD.
The equation for checking simultaneously the fatigue strength and static strength as well as the space required for the motion of TMD is given as follows:
where [r] and G are the allowable shearing stress and the shearing elasticity of the spring material respectively; r0 is the shearing fatigue limit of the spring material under pulsation shearing stress, nF is the safety factor for fatigue check; H0 is the original length of each spring; HB is the depth of the bridge deck; g is the peak factor; Ry, is the RMS displacement of TMD, K, C0, etc. are all coefficients of spring.
In Eq.(37), the two left items in brackets are both directly proportional to H0, while the third item is inversely proportional to it.
This means that when takes the maximum value, which can be bring the constraint Eq. (37) into full play. So, H0 can be appropriately determined from Eq.(38).
Conclusion:
(1) With the increase of span-length, the buffeting problem becomes more important compared with flutter phenomenon, which can be solved through some aerodynamic measures even for extra-long-span suspension bridges.
(2) Two basic buffeting theories developed by A. Davenport and R.H. Scanlan independently have been close each other by considering fully and rationally all important factors concerned.
(3) For the buffeting response analysis of long-span bridges, the approximate calculation formulae including some factors similar to the seismic response spectrum method can be established through the rational simplification to the theory of buffeting response spectrum.
(4) The buffeting-based selection of deck section is important in the preliminary design stage together with the conventional flutter-based selection. As a relative comparison, the slope of lift force curve of deck section is a predominant parameter in the buffeting-based selection.
(5) A method for the buffeting response in time-domain is useful in considering the non-linear factor both structurally and aerodynamically, as well as the effect of additional dynamic attack angle, which makes the buffeting analysis more to reality.
(6) The buffeting response of long-span bridges with strong enough torsional stiffness is mainly contributed by the vertical bending mode, and the TMD set is effective and convenient in suppressing vertical buffeting.