Developments in the Aeroelasticity of Bridges | Wind Engineering

In this article we will discuss about the developments in the aeroelasticity of bridges.

For over 100 years prior to the pivotal Tacoma Narrows bridge disaster of 1940 flexible bridges had been noticeably disturbed and some severely damaged by the wind. However, the late 1930’s and early 1940’s witnessed introduction of new consciousness and appreciation of what kinds of forces were at work in such events.

The relatively new field of aeroelasticity not yet known by that name had been developing since the late 1920’s in the aeronautical field. By 1935 the United States NACA had in hand the pioneering theoretical papers of Theodorsen and Garrick (1935, 1940) on air­foil flutter, and comparable theory was developed in Germany by Kussner (1936) and others.

When the Tacoma Narrows incident came along it was natural to attempt to infer some connection between bridge and airfoil flut­ter, which idea Bleich (1948, 1949) proceeded to exploit. Unfortunately Bleich’s effort was wide of the mark in that it implicitly attributed airfoil characteristics to a bridge deck cross section.

This easy but quite inap­propriate transposition of ideas stood as somewhat of a discouragement to sound analytical attempts at describing bridge flutter, since the latter, as compared to the airfoil variety, is in most cases a distinctly different, bluff- body phenomenon. Pugsley, in a commentary on Bleich’s work, suggested the possibility of measurement of motional, wind-induced forces. Halfman (1952), using an airfoil, demonstrated a method by which such measurement could be accomplished.

Initial analytical inadequacies were of course compensated for early-on by recourse to testing. Wind tunnel models of bridges particularly the repre­sentative deck-section model were early introduced from as early as 1940 by Farquharson (1949-54), Scruton (1952), Karman and Dunn and others.

Section models were extensively exploited by Steinman (1943-50, 1954), Selberg (1961), Kloppel, and by Walshe (1963). Later models took on a variety of partial and full-bridge forms. For a number of years now this strongly empirical approach has dominated the field.

Alongside it, however, has grown up a body of analytical theory that has begun to reduce this empiricism, explain some of the physical mechanisms at work, and exhibit good accuracy and productive capability. This theory coupled with appropriate basic experiment is presently proving quite valuable in the design studies of new flexible bridges.

The central points of departure of the theory around which its more recent greater degree of success has turned have been recognition of the following:

a) The value for design of the aerodynamic force coefficients;

b) The intrinsic relations, and differences, between static aerodynamic force coefficients and their dynamic (motion-related) counterparts (i.e., the “flut­ter derivatives”);

c) The need and means to make separate experimental determinations of each;

d) The need to draw sharp and clean distinctions between, on the one hand, the mechanical, and on the other, the aerodynamic aspects of any type of test model;

e) The need to distinguish between the dynamics and aerodynamics of a given model and those of the prototype bridge structure; and

f) The nature of the turbulence in the approach flow and its influence upon the distribution of forces over model or prototype.

A section model of a typical bridge deck section has classically been used to acquire static lift, drag and moment coefficients of the prototype. This was time-honored practice. On the other hand, the same model, allowed to move in proper degrees of freedom (vertical, torsion and sway), can lend insight into the sectional “flutter” forces.

Such forces, de­pendent on near-wake flow phenomena, cannot be inferred from steady-state conditions although they become asymptotic to the latter at zero frequency. Test methodology to measure such forces was unused for bridges until the mid-1960’s. It requires, for its exploitation, various sophisticated techniques of realizing model displacement and of implementing system identification to interpret the aerodynamic force results, which partially explains its rela­tive slowness in coming into common use.

Following Halfman’s airfoil stud­ies, the practice of measuring flutter derivatives was generally introduced to bridge applications, to the writer’s knowledge, by a paper of Ukeguchi, Sakata and Nishitani (1966). This was followed by papers by Scanlan and Sabzevari (1967, 1969) and by Scanlan and Tomko (1971), and since then by many others of increased sophistication, of which those by Szechenyi (1973, 1987), Larose, Davenport and King (1993), Raggett and Scanlan (1993), Sarkar, Jones and Scanlan (1992), Curami and Zasso (1993) are more re­cent representatives.

The notion of flutter derivative (as of the earlier-recognized static force coefficients) derives directly from the legacy of the aeroelastic theory of aeronautics, particularly the Theodorsen (1935) work. However, measuring bridge deck flutter derivatives began to reveal a special and quite typical trend among bridges one derivative in particular, the torsional aerody­namic damping derivative, consistently reversed sign with increasing (dimensionless) velocity, in contrast to the consistently stabilizing action of the analogous derivative for airfoils.

This outstanding effect characterized most bridge flutter as strongly driven by single-degree negative damping in torsion. This tendency has continued to typify bridge aerodynamics until recently when modern deck profiles have become more streamlined. Clas­sic coupled, vertical/torsion flutter of the airfoil type can occur in bridges, but the coupling effects, revealed via the “cross” flutter derivatives, are rel­atively weak. In certain respects most bridge flutter is closer to the stall flutter of airfoils than to its classical counterpart.

Although from the beginning in aeronautics flutter derivatives were con­sistently employed in the time domain to detect the critical (oscillatory) flutter condition, their proper interpretation within the frequency domain was not emphasized in the bridge context until much later.

The practice of using the sectional model uniquely as an analog source of both static force coefficients and of flutter derivatives and not as an (in­evitably improper) simulator of the full prototype bridge was a step finally acknowledging the kind of information correctly to be inferred from it.

When a section model is oscillated, the principal type of frequency-domain information to be extracted from it consists of the motion-related flutter derivatives. When, on the other hand, the frequencies of interest lie in the turbulence components of the approach flow, an alternate kind of frequency- domain information to be extracted from the model is the aerodynamic ad­mittance.

In the airfoil case the corresponding associated frequency-domain function is the Sean function. This complex lift function is related in a Fourier-transform manner to the Kussner vertical-gust-wall- penetration lift function of the time domain.

In the airfoil case, the time-domain lift-growth function which corresponds to an Inverse Fourier transform of the Theodorsen flutter circulation function is known as the Wagner function. The corresponding (but usually very differently-shaped) function for bridge decks was shown to be derivable from certain of the measured flutter derivatives.

Y.K. Lin and his co-workers have pursued exploitation of such indicial lift functions in the bridge context with the aim of observing, in time-domain studies, the effect of oncoming turbulence upon the flutter stability.

These studies to date, con­cerned with the basic question of how the stochastic nature of coefficients governs the stability of response, have been exclusively of a mathematical nature and have not benefitted from measurement of the subtle physical changes in aerodynamic forces that can be occasioned by turbulence in the oncoming flow.

Alternate experimental approaches have directly inferred flutter deriva­tives under turbulent approach flow, using identification techniques. Such methods have facilitated flutter analysis in the frequency domain. Typical results have shown sharper differences between smooth-flow and turbulent- flow flutter derivatives mainly in those cases where the bridge deck forms have received little or no aerodynamic treatment.

Turbulence in the approaching wind has been demonstrated to increase the velocity at which flutter occurs and, in general, the trend into instability of model bridges, making it more gradual. Testing has shown repeatedly that the wind speed of entrance into full instability for full-bridge models is higher and less abrupt when testing occurs under turbulent flow, as compared to section or full-bridge models of the same form under laminar approach flow.

This trend to higher instability velocities can be accounted for analytically by postulating a span-wise diminution of lateral coherence of the flutter or motion-induced forces, such as has analogously been measured in the case of vortex-induced forces.

Often in analyses for flutter, these reduction-of-correlation effects of motion- induced forces under turbulence can conservatively be neglected, to the end that oncoming turbulence is not counted upon during design as a stabil­ity enhancement, since the estimated level may not always be assured in the field.

However, no experimented cases have been reported wherein tur­bulence has occasioned actual flutter onset at a lower wind velocity than occurred under laminar approach flow. Clearly the coherence properties of force effects under smooth flow will inevitably be stronger than those under turbulent flow.

The primary preoccupation with motion-induced forces and their effect upon instability has been a step toward a state of the art where the natures of the phenomena involved are now quite well understood. Once a set of flutter derivatives has been measured for a given deck geometry, accurate and conservative prototype design forecasts are routinely being made with available modern analytic methods.

With the insight gained from examining the evolution of the flutter derivatives with dimensionless velocity, the geometric forms of several modern bridge decks have been strongly influenced toward wholly stable contours. However, an adjacent problem that of bridge buffeting response presently remains of strong concern, whether the deck form is intrinsically stable or not.

Conventional structural studies of bridges have encountered little difficulty in assessing conservative wind pressures for use in basic static design, but for fluctuating wind such as occurs due to mechanical stirring of the earth’s boundary layer under high winds its effects upon structural response have demanded more detailed analysis. First, the spectral nature of the natural atmospheric turbulence itself has required assessment.

Second, the manner in which the turbulent wind ‘seizes the structure’ has been a preoccupying concern. This phenomenon is itself basically an unsteady one (or at least a frequency-dependent one), its time- dependent variations arising from two factors the turbulent approach flow and the ‘signature’ distortion of the flow by the bridge itself.

Davenport (1962) was among the first to develop an analytic approach, based on steady-state aerodynamic theory, to the response of extended- span flexible bridges to gusty wind. In certain respects this employed con­cepts used by Liepmann (1952) for aircraft gust response analysis. Holmes (1975) also considered the problem, enlarging on Davenport’s treatment.

Early formulations of the problem were based upon static force coefficients; considerations of flutter derivatives were ignored, although ‘aerodynamic damping’ a feature later seen to be more explicitly incorporated via the flutter derivatives was empirically included from steady-state informa­tion.

Scanlan and Gade (1977) and Scanlan (1988) discussed the gust re­sponse problem, identifying aerodynamic damping via the flutter deriva­tives, though still employing a quasi-steady formulation for the gust forces, which was equivalent to assuming unit aerodynamic admittance.

An important consideration required by early works to bring calculation into line with experimental results was the modification of the analysis based upon steady force coefficients to a more realistic unsteady force format. This was empirically done by inclusion of the ‘aerodynamic admittance’ in the spectral response equations.

For want of better information, in the power spectral form of the problem the square of the absolute value of the airfoil-related Sears function was typically employed to this end, although Holmes commented on its shortcomings in this con­text.

Jancauskas and Melbourne (1986) later showed how near (or far) from the mark this device may come for some aerodynamic shapes. In a few cases it appears remarkably close; in others it is highly un-conservative. Appro­priate experimented evaluations of aerodynamic admittance have become an additional asset to bridge investigations.

The writer has delineated basic two-dimensional aspects of the combined stability and gust response problems using a frequency-domain approach, demonstrating the roles of flutter derivatives, the aerodynamic admittances, the implied relationships among them, and their links to an­tecedents in aeronautical aeroelasticity.

The (3D) lateral coherence charac­teristics of the natural, turbulent wind have been quite well documented. This information serves as a preliminary guide to the lateral coherence characteristics of buffeting forces over the span of a full bridge. Studies by Larose (1992) have revealed a somewhat greater degree of coherence among span-wise buffeting forces than estimated from atmospheric coherence alone.

The lateral coherence factor is one of the most important in analytical buffeting response estimations, as it indicates a very great net reduction in generalized force as compared to that calculated assuming fully coherent effects.