Non-Linear Effects in the Buffeting Problem | Wind Engineering

In this article we will discuss about the non-linear effects in the buffeting problem.

The classical approach to modelling the stochastic wind loading process and to calculating the structural response it induces relies on several linearizations. This applies particularly to the buffeting, along-wind and cross-wind, loads which are imposed by wind gustiness and by signature turbulence due to flow separation.

Linear models have also been developed for strongly non-linear phenomena such as flutter, galloping, and vortex resonance. These shall not be considered in what follows. The first linearization concerns the wind speed-wind load transfer. It is assumed that the load fluctuations are related linearly to the velocity fluctuations; higher order terms contained in the basic fluid dynamic equations are neglected. In other words, the aerodynamic admittance is modelled as a linear filter.

As well, the structural behaviour is represented by a linear, differential equation, in which stiffness and damping are constants. Additional interactive forces are caused by the displacement velocity of the structural movements, and are accounted for as linear aerodynamic damping and stiffness within the scope of the linearized theory.

The linear theory was developed, starting in the early sixties, by the fundamental work of Alan G Davenport. It provided for the first time a realistic design tool for wind effects on structures, and has become the backbone of modem wind loading codes.

The linearizations make the theory particularly efficient for two reasons. Firstly, the calculations can be performed in the frequency domain using Fourier Transforms and making use of a well-developed mechanical theory, the spectral approach. The second reason is that a system exposed to a Gaussian input typical for turbulent velocity fluctuations, will respond with a Gaussian output. This allows to predict statistically the extremes of the linear structural response.

On the other hand it is clear that most structures behave strongly non-linear in their ultimate limit state immediately before they fail. Modern design methods aim at calculating this state using higher order mechanical methods. It seems necessary to inspect whether the linear wind loading theory is applicable within this new type of design calculations and to review more appropriate methods.

Linearizations:

Elastic Non-Linear Restoring Forces:

Guyed masts respond to wind excitations with forced vibrations of the mast, forced and parametric excited vibrations of single ropes or bundles of ropes, and oscillations of the guy-mast system. The restoring forces of guy-mast systems are non-linear because of the sags of the tilted rope bundels.

The degree of non-linearity is determined by the amount of initial stress, the length, and the weight of a rope. The dynamic behaviour of a guy rope is governed by a non-linear differential equation of motion (DEQM), which contains square and cubic non-linear terms, and terms of parametric excitations.

Tonis (1989) tested spatial oscillations of a tilted rope, which undergoes a forced excitation by the periodical, longitudinal motion of its upper support, like it is imposed from a periodically moving mast. He stated that linearized DEQM cannot describe the spatial movement. The number of response frequencies of a rope to a mono-frequent forced excitation is drastically reduced. Cai and Chen (1994) found similar results for the forced excitation of the upper support in orthogonal directions.

However, guyed masts are often modelled as an elastically supported beam, which means that the guy supports are treated as linear. The linearization of the non-linear equation of motion is performed around the deflected position under average wind conditions or for the initial un-deflected state.

To reduce the computational effort, Gerstoft (1986) proposed a simplified procedure, in which bending moments of the mast are calculated as the background response using influence lines from the linearized system and applying patch loads to the mast. The dynamic response is estimated by multiplying the background response by a dynamical amplification factor.

This is possible, since the deformations, which result from applying specific patch loads, are in affinity to the important mode shapes. For design purposes, simplifying approximative equations are given in the mentioned paper to determine the amplification factor. The author limits the applicability of the procedure to certain ranges of mass and stiffness properties and ‘slackness’ ratios for the guys.

The latter ratio only treats the initial stress of the guys in the un-deflected position. Kahla (1994) pointed out that lowest oscillatory frequencies for the mast have been found for slack guys while the structure vibrates about its deflected position under average wind conditions.

Large oscillations of leeward guys are possible accompanied by large guy tension which likely would cause the collapse of the structure, either by rupture of guys or guy connectors, or by local buckling of some elements of the mast Peil (1993) proposes to linearize the restoring force around the operating point of the loaded guy- mast system.

The stiffness of the guy support depends on frequency, since the mass of the ropes is taken into account. The mean wind loads on the ropes are included into the computation of the operating point It is found that the mean wind loads have to be included for the computation of the support’s stiffness.

The calculated Eigen-frequencies and the spectral location of measured peaks are in good accordance, but the amplitude spectrum of e.g., the rip deflection is considerably overestimated. To improve the linear approximation, Peil (1993) adopts an equivalent damping ratio, which is determined from the peaks of the non-linearly and the linearly calculated amplitude spectra.

The difference of the amplitude spectra is explained in Petersen (1992) as a consequence of the permanent detuning of the stiffness of the support, which acts as a damper. A comparison of the amplitude spectra of a non-linear guyed mast and a linearized system is shown in Fig. 1, where the stiffness of the linear oscillator is the tangential stiffness of the original non-linear system in the un-deflected position, the damping ratio is the same in both cases It is stated in Petersen (1992) that the non-linearity leads to a jump phenomenon with a considerably reduced maximum amplitude, which is not affected from viscose damping.

Chaotic oscillations are detected. He found, that the response of a guyed mast in turbulent wind is limited to much smaller amplitudes as those of a linear system with the tangential stiffness of the guyed mast in un-deflected position.

Statistical equivalent linearization:

The equivalent linearization technique is an efficient analytical tool for approximating the response of systems with weak and in some cases even strong non-linearities, provided the structure is deterministic. A certain drawback of the method is that the probability distribution of the response must be known to some extend.

A DEQM of a linearized oscillator is constructed, which satisfies the non-linear response in a mean sense In the field of random vibrations, the method was first applied to non-linear SDOF-systems under stationary excitations The approach were later extended to MDOF-systems and to hysteretic structures.

Where M, D, K are the equivalent mass, damping, and stiffness matrices of the linearized system with the entries m_ij, d_ij, k_ij, i, j = 0,1,2,…,N, N is the number of DOF.

Atalik and Utku (1976) proved that the matrix entries satisfy the criterion eq. 3 for a Gaussian probability density function (PDF) of the response quantities. The derivation inside the brackets in eq. 5 means the calculation of a tangential mass, damping and stiffness properties from g(q̇ ̇, q̇, q) for each DOF.

After the ensemble average is performed, a single matrix entry contains the expected value of the tangent property, which is similar, but generally not the same as the partial tangent property at the (discrete) operating point. The mean and r.m.s response of eq. 2 is an approximation of the statistical response parameters of the original non-linear system and can be calculated by use of usual methods. The spectral method is advantageous for turbulent wind excitations.

In the case of a considerable deviation of the real non-Gaussian response from the Gaussian assumption, the accuracy of such statistical equivalent linearization is not adequate, since an inexact PDF in eq. 5 introduces an error into the first two moments. Moreover, a reliability analysis is very sensitive to the tail of the PDF of the response Pradlwarter et al. (1988) extended the approach to hysteretic MDOF-systems under non-stationary excitations and, as the essential point, to a response PDF of arbitrary shape.

Therefore, a vector transformation is established, t(q) can be looked at as a realization of the response vector of the non­linear response, and F_t is the distribution function of the non-linear response. The shape of F_t is constructed, e.g., by physical consideration based on prior knowledge, or by a Monte-Carlo-simulation over a small number of samples

The ensemble averages for the evaluation of the equivalent linearized system properties, like those in eq. 5, are obtained by integration of the transformed random variables t(q). It is proved in Pradlwarter (1990) that the mean and the r.m.s response of the linearized and the original non-linear system are identical.

As an application, the gust response factor (GRF) of concrete chimneys is investigated in Pradlwarter et al. (1991). The structure is subjected to loads which have a long-term character and to loads, such as wind turbulence, which are predominantly of short-term character.

The CICIND Model Code for Chimneys (1987) accounts for the viscoelastic behaviour of dynamically stressed, reinforced concrete by adopting a linear-elastic stress-strain relationship up to the ultimate limit stress of concrete, which is valid for the action of dynamic wind.

The ultimate limit stress f_ck/y_Cu for dynamic actions decreases with increasing long-term strain towards the long-term limit, but Young’s modulus remains constant for dynamic effects. The material law for reinforcement steel contains the elastic-ideal plastic behaviour of steel, the tension stiffening effect due to the bond between cracked concrete and steel bars is superposed in the stress-strain relationship P-A-effects are included into the analysis. Linear damping is assumed.

The non-linearity of the concrete chimney contains the softening of the structure caused by cracks which are assumed to develop due to gust action and not from fatigue. The wind loads are calculated based on the linearized quasi-stationary theory (cf. to 3.3). The GRF’s are determined for the stationary states of the structural oscillations. It has been found out that the GRF’s near the ultimate limit state are up to 20 % larger in the non-linear analysis than the respective values based on linear dynamic analysis.

This effect is explained by the ‘softening’ of the structure when it enters the highly non-linear range The Eigen-frequencies then decrease and move towards the frequency of the maximum ordinate of the load spectrum, which lets the response increase (Fig. 2).

Hysteretic Response:

A nominally elastic construction material, such as construction steel, follows Hooke’s law, when it is loaded within its elastic range and if the loading is slow so that the material can establish the deformed state proportional to the load intensity (quasi-static-loading).

When the stress is imposed or released as a step function the strain lags behind its elastic value (creep). In the case of a harmonic load, such behaviour becomes apparent in the time domain as a phase lag of the strain behind the stress. The stress-strain relationship shows a dynamic hysteresis loop which is at small stress amplitudes slim and elliptical in shape.

The term linear damping refers to this type of dynamic hysteresis. It can be modeled as the result of the combination of a linear spring and a viscous dashpot. The elastic part of the total energy, fed to the system, can be restored.

The dissipated part is usually represented as the loss factor Ψ of a material or construction component:

Are the rate of energy loss per unit volume and per loading cycle (w_dis), and the rate of specific deformation energy (w_def) per unit volume and cycle respectively, σ(ε) is the stress due a strain ε, E is Young’s modulus, and ε₀ is the maximum strain the material is subjected to.

When the stress amplitude of the harmonic load exceeds a certain limit even within the elastic range e.g., 1/20 of the fatigue strength of construction steel – experimental – hysteresis loops show a lancet-like shape, indicating a non-linear damping. The loss factor Ψ tends to increase with increasing strains. It is often assumed based on experimental results that Ψ increases proportional to the square of the deformations.

Petersen (1971) proposed to retain the linear model, applying an equivalent loss-factor determined from the relation for stationary oscillations.

The non-linear relationship eq. 8 just controls the lateral width of the loop, the index r stands for reference conditions, i.e., a reference strain amplitude So, and a corresponding loss factor Ψ_r obtained e.g., from an experiment That means that the hysteretic loops do not retain affinity with increasing ε₀, as it is the case for strictly linear oscillations, but become fatter.

The differential equation of motion can then be written as:

Where ω₀ is the Eigen-frequency for zero damping, q is the generalized displacement, q₀ is the (stationary) amplitude, q_0r is the reference amplitude tor determining Ψ_r, and Q/m is the generalized load amplitude normalized by the mass m.

The stationary solution for the amplification function V is given as:

Where q_st is the displacement under static load application.

The resonant behaviour is of major interest. Fig. 4 illustrates the change of the resonant response variances R̃² of a weakly damped steel structure (i.e., Ψ_r < 0.4) under buffeting wind load for damping proportional to the squared strain amplitude (R̃²) against the strictly linear approach (R̃²), following the procedure in Niemann (1990) for convenience.

The ratio q_0r/q_st can be determined a-prion by static methods and is set to 1.5 here. Values for V(ω/ω₀=1)/V₀(ω/ω₀=1) are taken from Petersen (1971), where V₀ is the amplification factor for Ψ = Ψ_r. It is obvious, that consideration of non­linear damping characteristics diminishes drastically the overall dynamic response of flexible steel structures under buffeting loads.

A linear, viscous model is no longer appropriate when the construction material is loaded at very high levels or beyond its elastic limit. The hysteresis loops become fat and sharp edged and can degenerate into shapes similar to those for Coulomb frictional damping. Fat hysteric loops occur also, when composite materials, such as reinforced concrete, is exposed to cyclic loading. For such cases Bouc (1967) and Wen (1976) proposed a model, in which the hysteretic force z is governed by the differential relationship.

where γ, v, A, n are parameters which control the shape of the loop A wide variety of hysteretic constitutive laws which can be expressed in this manner as was shown by Suzuki and Minai(1987). Such differential model of hysteresis can be combined with the system’s differential equations to yield an overall differential model, which can be analysed by using techniques such as statistical equivalent linearization.

The latter approach is applied extensively in earthquake engineering. The application of the method and of other stochastic approaches on hysteretic structures is reviewed in Roberts and Spanos (1991) and Schueller (1991). Only few works exist about the treatment of wind excitated structures in the highly non-linear range, which is dominated by restoring forces with fat hysteretic loops, such as degraded structures Chen and Ahmadi (1992) studied the sensitivity of the displacements of a base isolated three-storeyed building to wind loading.

A quasi- stationary model was used for modeling the longitudinal turbulent wind action. The variation of the wind speed with height was neglected. The building is considered as a rigid body. The results mainly may serve for the comparison of the effect of different isolation types in severe storms and hurricane conditions.

Procedures in the Time Domain:

General aspects:

Ultimate limit states of structures usually include large displacements and strong non-linearities of the structural behaviour. Parametric excitations and stochastic uncertainties in the structural parameters can be included. Nowadays, computer power allows for calculations in the time domain, even if the excitation is multi­dimensional and stochastic, such as the wind loading process. An important feature is, that flow-structure interaction and buffeting response are treated simultaneously. The calculations can be extended over a sufficient time interval in order to ensure the statistical relevance of the results.

In engineering practice, non-linear systems are mostly solved incrementally Starting point is the non-linear condition of equilibrium (t), g(q̇ ̇, q̇,q)= f(t), where g(q̇ ̇, q̇,q) contains all structural properties and the load function f(t), both the purely aerodynamic forces, depending only on time, and aeroelastic components, self-excited by the current structural response.

A suitable incremental iteration algorithm is presented in detail in Beem et al (1994) The FE-code FEMAS is applicable to physically and geometrically non-linear truss, beam and shell structures, arbitrary stress-strain relations can be treated.

Load vector processes f(t) are required which can be applied either as nodal forces or nodally-defined element loads The load vector processes may be obtained from measurements in full scale experiments or in a boundary layer wind tunnel, or ,more universally, from mathematical transformations of artificial wind time histories into loads, or direct generation of load processes.

Wind Speed Records:

For time-history calculations, the loading process may either be obtained from direct load simulations, or by transforming time histories of the wind speeds into loads through an adequate wind load model. One of these is the quasi-stationary model in which is assumed that the stationary aerodynamic coefficients are valid for the fluctuating loads.

Although the model is not very meaningful beyond some frequency limit, it has found wider application, and surprisingly good results can be achieved One of its merits is, that the flow-structure interaction can be included principally revealing linear and non-linear contributions of aerodynamic damping and stiffness.

More sophisticated models are based on the aerodynamic force response produced by a step change of the wind velocity, e.g., Kϋssner and Wagner functions However, the first step in any of these approaches is to provide time- histories of the wind speed.

The application of measured wind time histories in full scale experiments suffers from the high experimental effort for recording long enough multi-dimensional wind vector processes at a dense mesh of monitoring points. Moreover, such wind speed records include e.g., instationarities, which occur as trends without any visible period on the one hand. Sudden changes of the frequency content cannot be countered with trend removal techniques on the other hand.

However, natural wind speed records have been transformed to load histories. An advantageous method to collect wind time histories is the construction of a multivariate random field Basic generation techniques simulate the fluctuations of a single velocity component by superposing N cosinusoids per frequency f, N is the order of the model, each weighted with respect to the spectral ordinate of the target spectral density function.

For multiple wind velocity histories, the Shinozuka II method yields the requested number of velocity processes, which match the target cross-spectral density functions. Numerically optimized versions of such methods are successfully applied for the analysis of the load-bearing capacity of large structures under buffeting excitation.

It is generally accepted that the probability distribution of turbulence is Gaussian. While the target spectrum is usually well represented within the procedure mentioned above, it cannot be taken for granted that higher moments than of second match the Gaussian values. The statistical inaccuracy may be misleading in estimating the fatigue damage, since it generally varies in accordance with higher moments.

The filtering of Gaussian white noise, e.g., by an autoregressive (AR) filter, lead to more accurate statistical properties of the load AR methods have been adopted for the simulation of wind turbulence (Iwatani (1982)) Extended autoregressive filter types have been applied, e.g., ARMA (Maeda and Makino (1992)), for generating wind velocity fields.

The wind vector u(t), = u₁(t), u₂(t), …, u_2k (t)]^T may consist of the fluctuating parts of the longitudinal u₁(t), u₂(t), …, u_k(t) and orthogonal components u_k+1(t), u_k+2(t), …, u_2k (t), 1=1,2,…, k are the identifiers of the points at which the processes are required These are usually the structural nodes on which aerodynamic loads are imposed.

A general 2k-dimensional AR-model is given by:

Where z(t) =[z₁(t), Z₂(t), …., Z_2k(t]^T is the vector of independent Gaussian white noise processes. A(m) is a 2k x 2k-matrix comprising the coefficients of the AR- model which can be calculated solving the Yule-Walker-equations.

C(τ) is the matrix of covariance functions (CVM), τ is the time lag, M is the depth of regression of the AR-model, and D(τ) the (diagonal) covariance matrix (CVM) of the white noise processes. When the longitudinal u₁ = u₁‘, 1 = 1, 2,…,k and the vertical U_k+1= w₁ components of turbulent wind are considered, C(τ) is not banded as a CVM of the u’ and lateral v’ components is, since it follows from the condition of continuity that the u’ and w’ components are coherent signals. Suitable models for the coherence can be found in Kristensen et al. (1989).

The model-order selection process is rather empirical, since criteria, such as the Akaike-FPE, PAIC, BIC, or AIC unfortunately give contrary predictions. The natural frequencies of linear structures are often found beyond the spectral peak in the inertial subrange. In such cases, model orders of about 3 to 5 can be employed in structural analysis.

The interesting frequency range of non-linear vibration cannot be limited to some range around the Eigen-frequencies of the linearized system, the spectral properties of the AR-model should rather match the target cross-spectral density functions down to low frequencies.

Higher orders are required then for the basic AR-model, since autoregressive methods suffer from inconvenience for the representation of peaks. Found an excellent fit to the production zone, the peak range, and the inertial subrange of v. Karman target spectra at an order of 90.

Effect of non-linear contributions to the dynamic pressure:

When the buffeting forces are calculated using a quasi-stationary approach, linearizations of the mathematical model are not requested, since the dynamic pressure q(t) can be computed explicitly by q(t) = ρ/2 [(u+u’(t)²+v’²(t)]…(14).

ρ is the air density Eq. 14 yields a non-Gaussian dynamic pressure, since it is non-linear in u’ and v’, the wind speeds are usually supposed to be Gaussian distributed In fig 4 the non- Gaussian shape of the probability density function (PDF) of q after e.q. 14 is compared to a linearized approximation, for which the square terms of turbulence are neglected In the example, the turbulence intensity is 0.172, and Iu/Iv=0.8 is assumed.

The exceedance probability of an extreme value of the load increases with the right-sided skewness of the PDF of q. The non-normality of the PDF is specified by the skewness coefficient y₁ and the kurtosis y₂. These statistical parameters are defined in the table Neglecting lateral turbulence, the coefficient of variation (c.o.v ) of q is 0.3368, the skewness is 0.3368, and the kurtosis is 0.3442 for the example mentioned above.

Peinelt (1989) considers the response x of a linear SDOF oscillator under a non- Gaussian load due to e.q. 3 .3-1. The Eigen-frequency of the oscillator is 1 rad/s. The damping ratio is stepwise decreased from 0.09 to 0.01. The variation of damping is found to alter the higher statistical moments (cf. table). For zero damping the response approaches Gaussian properties.

The exceedance probabilities P_c of multiples of the r.m.s value of the response are compared in fig 5 Curve (a) denotes the P_c due to the non-linear force, curve (b) represents the exceedance probabilities of a Gaussian load of same c.o.v. The Gaussian excitation yields considerably smaller P_c for thresholds larger than three times σ_x.

Non-Linear Dependence of the Aerodynamic Coefficients on the Angle of Wind Attack:

In the frame of the classical quasi-stationary approach an aerodynamic coefficient c is solely a function of the angle of attack of the flow. It is obtained from measurements in smooth flow. The non-linear relation between the coefficient and the angle of wind attack must be known explicitly, in order to provide a function value c(ϕ) for each ϕ within an interesting range.

The precise mathematical incorporation of tabulated pairs of points c-ϕ into a formula presents little difficulties, e.g., for the representation of a curve with sharp peaks Singular points of discontinuity in smooth flow set a limit to the general acceptance of the representation as a Taylor-type series, since the differential quotient dc/dϕ is not defined in any cases.

Furthermore, the fluctuations of the angles of attack of the flow typically cover a range of about ±20°, so that (dc/dϕ)^.ϕ would result in significantly erroneous estimates of the actual coefficients. The consideration of the fluctuation Δ(ϕ'(t)) instead of (dc/dϕ)^.ϕ’ is a more exact description. Δ(ϕ'(t)) is the deviation of a momentary coefficient c(ϕ̅ + ϕ'(t)) from c(ϕ̅). To Δ(ϕ'(t)) belongs the PDF f_Δ(Δ).

The statistical properties of the non­linear transformed coefficients can be approximated without performing a Monte-Carlo-Simulation. The dynamic pressure and the angle of attack of the flow are assumed to be uncorrected for this approximation. One obtains f_Δ(Δ) from a nonlinear transformation of f_ϕ(ϕ’) considering the nonlinear interrelation between Δ and ϕ’.

The inverse function ϕ’= g^-1(Δ) may be ambiguous, so that here n is the number of ambiguities at Δ and |dϕ’/dΔ|is the jacobian, here for the case of a scalar transformation Δ̅ may be computed as the mean of f_Δ(Δ).

Then, the mean forces are given by:

Experimental verifications show, that even small scales of turbulence shift a mean value (fig 7). That can be modeled correctly, even for low- and high-turbulent flows.

Flow – Structure Interactions:

Stationary aerodynamic coefficients:

The general aeroelastic response of buildings vibrating in turbulent wind is described by g(q̇ ̇, q̇, q) = f(q̇ ̇, q̇, q, t)…(17)

The presence of response quantities on the right side of eq. 17 represents aeroelastic feedback, rendering even a linear structure to be non-linear, f contains both buffeting and aeroelastic components.

Forces induced by motion play an important role for the dynamic excitation of flexible, light structures, such as masts and chimneys long span horizontal structures, especially stayed cable bridges in the erection state, and large span roofs.

One class of force models that has been applied widely to structures both in smooth and turbulent flow is that based on quasi-stationary theory. The prime feature of that model is the assumption that the external forces acting on a body are not influenced by motion but that motion induces a further force component due to the actual state of deformation of the structure.

In the case of a line-like structures, interaction effects are described on a cross section with three degrees of freedom These are the translational displacements in mean wind, x(t), and lateral directions, v(t), and the torsional rotation, θ(t). The overall wind forces arise from a relative flow around the vibrating structure determined from relative velocities.

U_rel(t) = u̅ + u'(t)-ẋ(t), v_rel(t) = v'(t)-y(t) …. (18.b)

ẋ, ẏ are the time derivatives of the translational displacements x(t) and y(t). The lateral response velocity affects the angle of attack of the wind.

The current angle of attack of the flow is:

The calculation of an overall wind force acting on the relative flow direction follows:

Where the relative dynamic pressure q_rel depends on the relative velocity between the incident flow and the structure. A is the reference area, which is b, or b² respectively for the aerodynamic torsional moment, and c(γ(t)) stands for an aerodynamic coefficient, which is solely a function of the angle of flow attack in frame of quasi-stationarity.

In connexion with the treatment of structures in the time domain, there is no need to separate between external and motion-correlated components, as long as the right side of eq. 17 contains the buffeting forces and the aeroelastic forces after eq. 20. Such formulation can be characterized as a fully aeroelastic approach It allows an assessment of the stability of the structural response, yet not the structure itself.

The latter problem can be dealt with when the aeroelastic force contributions in eq. 17 are linearized and expressed as aeroelastic matrices multiplied by the vector of structural displacements (vector of aerodynamic stiffness forces) and its time derivatives (vectors of viscous aerodynamic damping forces or aerodynamic mass forces) Numerical problems can arise due to the fact that the tangent aeroelastic matrices are non-symmetric, unlike the tangent mass, damping and stiffness matrices of the linearized vector of restoring force g.

Aerodynamic damping is the significant aeroelastic component which should be accounted for in the buffeting problem of line-like structures. Approximative estimates are usually written as damping ratios and are derived by linearizing eq. 20 and separating the total force into buffeting and motion-correlated components.

Eq.s 21a, b contain the time-average values of the aerodynamic damping ratios. The lateral aerodynamic damping contributions dc_I/dϕ can add to and substract from structural damping. That may lead to negative total damping, which then cause large amplitude oscillations and in the extreme case lead to aerodynamic instability.

Non-Stationary Buffeting Coefficients:

Beyond some frequency limit quasi-stationarity is an inadequate representation of the variation of forces in time. The instationarities in the aerodynamic coefficients can be accounted for by indicial response functions. Indicial response functions incorporate the effects of time lag in development of aerodynamic forces behind the time histories u'(t), w'(t).

In Scanlan (1993b) the indicial response functions are presented for drag, lift and aerodynamic moment on bridge decks and linked to the Wagner and the Kϋssner functions. The Wagner function (cf. fig. 8) accounts for the lag of lift on a flat plate initiated by a step function change of the angle of flow attack, but it is supposed that it also can treat a step function change u'(s). A vertical gust w'(s) and, analogously, a vertical displacement z(s) of the bridge deck requires a Kϋssner-type function. The approach works in both the frequency and the time domain.

Conclusions:

The presented review gives an overview about the buffeting analysis of non-linear structures in the ultimate limit state, where structural non-linearities are dominant Contributions coming from structure-flow interactions are also considered.

The following conclusion can be drawn:

i. In general, the linear approach ensures a conservative design since the realistic, i.e., non-linear behaviour gives rise to additional load bearing capacities. However, this is not always true since softening of the structure may shift the dominant frequencies in the low frequency, high energy range of the load spectral density.

ii. Equivalent linearization is an attempt to maintain the benefits of the linear theory and at the same time to include the non-linear structural properties in terms of tangent restoring forces.

iii. Time history calculations using incremental-iterative algorithms can account for arbitrary non-linearities of restoring forces, interactive external forces. The buffeting loads may in this case be represented by AR or other simulation procedures in the time domain with good accuracy. Time domain approaches are therefore becoming increasingly important.