VIRTUAL WIND TUNNEL: AN ALTERNATIVE APPROACH FOR THE ANALYSIS OF BRIDGE BEHAVIOUR UNDER WIND EFFECTS
ABSTRACT
The study of bridge responses under wind-induced loads is based upon full aeroelastic model testing or hybrid methods which use section model tests and subsequent computer analysis. Both methodologies present several strong points and some shortcomings, specially related with the visualization of the bridge dynamic behaviour. Nowadays, advances and improvements in computational power and computer aided design technologies make possible a new way towards the feasible design of long span bridges considering its aerodynamic and aeroelastic behaviour. The virtual wind tunnel (VWT) technique developed by the authors joins together accurate section model testing with computer aided design in order to obtain a detailed computer visualization of the complete bridge behaviour under wind flow. The results obtained for the TacomaNarrowsBridgeis presented.
CONTENTS
1. Introduction(3)
2. Flutter phenomenon in bridges(5)
3. Hybrid method for the evaluation of wind-dependent static loads and flutter speed in bridges
3.1. Hybrid method for the evaluation of wind-dependent static loads on bridge decks (7)
3.2. Hybrid method for the evaluation of the flutter speed in bridges (9)
4. Tests of full bridge models in boundary layer wind tunnel(15)
5. Comparative study of available methods for the analysis of the aeroelastic performance of bridges(17)
6. Virtual wind tunnel: concept and achievements (19)
6.1.The Tachoma Narrows Bridge And Its Failure As Explained By VWT method(23)
7. Conclusions (28)
8. References (29)
1. INTRODUCTION
Suspension and cable-stayed bridges are wind prone structures which require detailed and complex studies in order to guarantee its safe behaviour under wind. Several wind-induced vibrations have been described in bridge technical literature, although some of those types are more critical or probable than others. In fact, one of the most important aeroelastic instabilities is the flutter, as it can be responsible for the complete destruction of a bridge, as it was the case with the TacomaNarrowsBridge. On the other hand, prevention against flutter phenomenon determines the fundamental design characteristics of long span bridges.
Therefore, a lot of work and research has been done since November 1940, when the aforementioned TacomaNarrowsBridge collapsed, as in those days aerodynamic performance of structures was a newborn subject and its study was reduced mainly to the aeronautic realm.
In the civil engineering field wind effects on structures were first treated by means of experimental studies using boundary layer wind tunnels. As time passed by, the out coming of new technologies allowed the development of new techniques in experimental testing, for instance the study of section models, as well as the incorporation of computational methods for the analysis of bridge performance under wind flows. In this dynamic and ever changing environment hybrid methods (combination of experimental and computational techniques) for the study of aerodynamic and aeroelastic phenomena appeared.
Nowadays, advances and improvements in computational power and computer aided design technologies make possible a new pace in the way towards a feasible design process of bridges considering its aerodynamic and aeroelastic behaviour. In this paper are going to be presented the results obtained when the best of two worlds is joined together: accurate experimental testing and computer aided design in order to bring out what is going to be named as virtual wind tunnel (VWT). This VWT allows engineers to get a detailed visualization of the complete bridge behaviour under wind flow while some of the shortcomings and expenses of full bridge aeroelastic models are avoided.
2. AEROELASTIC FLUTTER
Fluttering is a physical phenomenon in which several degrees of freedom of a structure become coupled in an unstable oscillation driven by the wind. This movement inserts energy to the bridge during each cycle so that it neutralizes the natural damping of the structure, thus the composed system (bridge-fluid) behaves as if it had an effective negative damping (or had positive feedback), leading to a exponentially growing response; in other words, the oscillations increase in amplitude with each cycle because the wind pumps in more energy than the flexing of the structure can dissipate, and finally drives the bridge toward failure due to excessive deflection and stresses. The wind speed which causes the beginning of the fluttering phenomenon (when the effective damping becomes zero) is known as the flutter velocity. Fluttering occurs even in low velocity winds with steady flow. Hence, bridge design must ensure that flutter velocity will be higher than the maximum mean wind speed present at the site.
Flutter is a self-starting and potentially destructive vibration where aerodynamic forces on an object couple with a structure's natural mode of vibration to produce rapid periodic motion. Flutter can occur in any object within a strong fluid flow, under the conditions that a positive feedback occurs between the structure's natural vibration and the aerodynamic forces. That is, that the vibrational movement of the object increases an aerodynamic load which in turn drives the object to move further. If the energy during the period of aerodynamic excitation is larger than the natural damping of the system, the level of vibration will increase. The vibration levels can thus build up and are only limited when the aerodynamic or mechanical damping of the object match the energy input, this often results in large amplitudes and can lead to rapid failure. Because of this, structures exposed to aerodynamic forces - including wings, aerofoils, but also chimneys and bridges - are designed carefully within known parameters to avoid flutter. In complex structures where both the aerodynamics and the mechanical properties of the structure are not fully understood flutter can only be discounted through detailed testing. Even changing the mass distribution of an aircraft or the stiffness of one component can induce flutter in an apparently unrelated aerodynamic component. At its mildest this can appear as a "buzz" in the aircraft structure, but at its most violent it can develop uncontrollably with great speed and cause serious damage to or the destruction of the aircraft. Flutter can be prevented by using an automatic control system to limit structural vibration.
Flutter can also occur on structures other than aircraft. One famous example of flutter phenomena is the collapse of Galloping Gertie, the original TacomaNarrowsBridge.
Fig 1. Flutter of the TacomaNarrowsBridge
3. HYBRID METHOD FOR THE EVALUATION OF WIND-DEPENDENT STATIC LOADS AND FLUTTER SPEED IN BRIDGES
3.1. Hybrid Method For The Evaluation Of Wind-dependent Static Loads On Bridge Decks
The static load caused by the wind pressure acting on a bridge deck can be obtained by means of a hybrid method. This method is well established in wind engineering [2] and [3] and it begins with an experimental phase that must be completed in order to obtain the deck aerodynamic coefficients, which depend upon the angle of attack α between the deck and the oncoming wind flow.
Fig 2.
In FIG a scheme of the lift, drag and moment aerodynamic forces acting on the deck is shown.
These tests must be carried out with the section model being fixed at different angles of attack while the loads due to the oncoming flow are measured
Drag aerodynamic coefficient vs. angle of attack for the GreatBeltBridge
The second phase in this methodology consists of the computational evaluation of the static forces acting along the bridge deck. A finite element model of the studied bridge must have been worked out (see Fig. 5) and the static loads can be evaluated using the following expressions:
(1)
where D is the drag force per unit of length, L is the lift force per unit of length, M is the moment per unit of length, ρ is the air density, U is the wind speed, B is the deck width and CD, CL and CM are the aerodynamic coefficients obtained in the wind tunnel.
Fig 3. Finite element model of the GreatBeltBridge.
3.2. Hybrid Method For The Evaluation Of The Flutter Speed In Bridges
The foundations of this method for solving the flutter problem were established by Scanlan and Tomko in 1971, although new developments were published by several researchers during the following years, until the present time. Analogously to the former case, two different phases must be completed in order to evaluate the flutter wind speed in bridges. The first task to be carried out is the experimental measurement of the flutter derivatives, also called Scanlan derivatives, using a section model that can be undergoing free oscillations or subjected to forced oscillations inside the wind tunnel test chamber.
Fig 4.a 4.b
a)Principle structural setup of a suspension bridge. b) Cross section of bridge deck
Up to 18 flutter derivatives can be obtained which depend upon the reduced frequency K=Bω/U, where ω denotes circular frequency. In Fig. 5 an example of the flutter derivatives of the GreatBeltBridge plotted vs. the reduced frequency is presented.
Fig 5. Flutter derivatives to for the GreatBeltBridge.
View Within Article
The second step consists in the numerical evaluation of the aeroelastic forces acting on the bridge deck .
Fig 6. Evolution of aeroelastic damping vs. wind speed.
4. TESTS OF FULL BRIDGE MODELS IN BOUNDARY LAYER WIND TUNNEL
The aim of this technique is to reproduce in the laboratory the aeroelastic behaviour of the prototype. Therefore a model that replicates the future structure must be built and immersed in an oncoming air flow. Several responses can be obtained from these tests such as reactions, deflections under wind load or unstable behaviour at certain wind speed caused by flutter.
The first full aeroelastic test of a bridge was that of the TacomaNarrowsBridge in the 1940s used to investigate its collapse. Experimental techniques have evolved a lot, until the present days when this keeps on being an active research field. Model scales common in full bridge modelling are about 1:100–1:500, depending upon several factors such as wind tunnel dimensions or similarity requirements.
View Within Article
Nowadays a carefully planed experimental campaign must include the following tasks:
In first place, a section model test must be carried out on a relatively large model scale (1:100 or even 1:25) in order to determine the deck aerodynamic behaviour. Then, a second section model test must be performed using a model scale equal to the bridge full model scale to be adopted. This second section model should be modified until it shows an aerodynamic behaviour equivalent to that of the first section model. Once this condition has been satisfied, the full aeroelastic model must be constructed replicating the characteristics of the second section model which was built with the same model scale. Therefore, full aeroelastic model must include previous section model tests in order to guarantee a good model performance.
5.COMPARATIVE STUDY OF AVAILABLE MEHODS FOR THE ANALYSIS OF THE AEROELASTIC PERFORMANCE OF BRIDGES
This section is going to focus on the hybrid method and full model testing for study bridge aeroelastic performance. Alternative approaches such as those based upon computational fluid dynamics (CFD) are not considered due to the limited results obtained to date, although continual progress is being made in this field.
The main advantages of section model tests used in the hybrid method can be summarized in the following:
• Relatively low cost of sectional models themselves and the wind tunnel facilities.
• The model scale must be large, about 1:25–1:100, although exceptions can be found in literature. This allows proper modelling of important geometric details as well as reducing possible distortions due to Reynolds number effects.
• Section model tests can be carried out in small size wind tunnel facilities.
• Both geometric and dynamic model properties can be modified easily.
The main shortcoming that can be found is
• Standard section model techniques can offer inaccurate results due to three-dimensional effects of topography or deck geometry.
The full aeroelastic model technique presents several strong points which are [9]
• Full aerodynamic interaction between deck, towers, abutments and cables can be modelled.
• Wind characteristics across the span can be obtained when a model of the topography is also included.
• A complete set of aerodynamic responses can be obtained, such as reactions, displacements or aeroelastic instabilities.
• A clear visualization of the model deflections under wind flow can be obtained. However, due to similarity requirements the oscillation frequencies are higher than the correspondent ones in the prototype. In fact, when Froude scaling is respected, the frequency scale is equal to the inverse of the square root of the length scale, for instance, for a length scale of 1/100 the frequency scale λn is 10, therefore, for this considered example, full model oscillations are going to be 10 times faster than the real ones in the prototype. This circumstance darkens the perception of the real bridge dynamic behaviour under wind flow.
Additional weak points of this method are listed below:
• High cost of both boundary layer wind tunnel facilities and the full aeroelastic models to be used. In addition, section model tests must be carried out in order to ensure the experiment reliability.
• It is difficult to introduce modifications in full models if their aerodynamic behaviour is inadequate.
• Due to the existing trend of building bridges with longer spans each time, the size of wind tunnels must also be increased in order to maintain adequate model scales.
Both methods, hybrid method and full model testing can be used to identify the wind flutter speed, as the two usually offer close results.
6. VIRTUAL WIND TUNNEL:CONCEPT AND ACHIEVEMENTS
What do wind engineers dream about? For the authors’ envision the answer to the former question is to be able to anticipate the real structural behaviour of long span bridges under wind flow. The virtual wind tunnel (VWT) is the tool that can turn dreams in facts. The VWT applies the hybrid method, explained in previous sections, to evaluate the bridge response to an oncoming wind flow and additionally produces a realistic animation of the bridge behaviour by means of a digital visualization model. Therefore the real deflection of the bridge can be obtained and realistically reproduced for a wind speed range between zero and the flutter speed. Two different situations must be considered.
For a uniform wind speed lower than the critical flutter speed, the VWT obtains the static deflection of the bridge under wind load. The structural problem to be solved is
Ku=p(U),
where p(U) is a vector containing the aerodynamic wind loads, which depend upon the aerodynamic coefficients obtained using a conventional wind tunnel and the flow speed. Wind loads are different for different wind speeds, therefore the VWT is able to simulate the changes in the static bridge deflection for a wind speed increment form U=0 up to any speed UUf in a time interval t.
The second phase to be considered is the one that corresponds to U=Uf as this is the critical wind speed for flutter. In this case the bridge deflection is obtained throughout the eigenvector problem defined in (5), which leads to the following expression for the time-dependent bridge deck deflection:
u(t)=Φwjeμjt=Φwjeiβjt,
where Φ is the modal matrix, and subindex j corresponds to the jth aeroelastic mode which satisfies αj=0. Eq. (9) gives the bridge deck movements as a function of time for the situation of neutrally stable motion previous to the instable state associated with flutter. The VWT reproduces that steady harmonic oscillation, without frequency scaling, which is added to the bridge previously evaluated static deflections caused by the wind aerodynamic load by means of a realistic visualization computational model.
A frame from the animation of the MessinaStraitBridge deflections under aerodynamic loads caused by a wind speed lower than the critical is shown.
Fig 7.Static deformation of the MessinaStraitBridge under static wind load.
Four frames from the digital animation of the steady state oscillation plus the static deflection caused by the aerodynamic wind loads of the MessinaBridge when the flutter speed is reached are presented
Fig 8.Realistic frames of the aeroelastic response of the MessinaStraitBridge for critical wind speed.
The VWT was applied for the first time to the collapsed Tacoma Narrows Bridge as a movie showing the vibrations that lead to the failure existed and the computer animation produced using this technology could be compared to identify the similarity. This was a way to pay tribute to the bridge that opened the way for the wind engineering science. In fig 9,picture of the real oscillations of the TacomaBridge and a frame from the animation of the TacomaBridge performance under wind flow obtained using VWT techniques are shown.
Fig 9.a 9.b