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The Eighth Asia-Pacific Conference on Wind Engineering,December 1014, 2013, Chennai, India

NON-FLUTTER DESIGN PRINCIPLE FOR LONG SPAN BRIDGES Jens Johannson1 Michael Styrk Andersen2DQGMichele Starch vre2

1 Assistant professor, Ph.D., University of Southern Denmark, Faculty of Engineering, [email protected] 2Student M.Sc. Eng., University of Southern Denmark, Faculty of Engineering

ABSTRACT The case of flutter on a sharp edged flat plate section model, with a height-to-width ratio of 1:10, has been investigated at four different torsional-to-vertical frequency ratios equal to 0.71, 0.88, 1.19 and 2.10. At a torsional-to-vertical frequency ratio of approximately 1.1 the flutter wind velocity for a thin airfoilshows an asymptotical behavior. In traditional bridge design the torsional-to-vertical frequency ratio is increased to obtain higher flutter wind velocities. In the present study, we investigate, what we will label the non-flutter design principle, in which the torsional-to-vertical frequency ratio will deliberately be smaller than 1. Two cases with frequency ratios of 1.19 and 2.10 showed classical and torsional flutter, respectively.Within the maximum wind velocity obtainable in the wind tunnel the two cases with a frequency ratio below 1 did not show any sign of aerodynamic instability. The results are a first indication the non-flutter design principle could be applicable in future design of slender structures.

Keywords: Wind induced vibrations, frequency ratio, non-flutter design principle, flat plate

Introduction The catastrophic failure of the Tacoma Narrows Bridge on November 7th 1940 has led to

bridge design procedures that naturally include the investigation of the aeroelastic phenomenons including flutter. Classical flutter is a dynamic instability, where vertical and torsional modes couple, at a specific wind speed, causing huge vertical and angular displacements and ultimately possible structural failure.Theodorsen (1934) formulated a general theory of the flutter mechanism based on the flow around a thin airfoil. His theory has later been adopted for the case of bridge flutter and been the subject to a significant amount of research. See e.g. the textbooks of Simiu and Scanlan (1996) or Dyrbye and Hansen (1996) of which we shall adopt the formulation.

Figure 1 shows the flat plate. The two degrees of freedom; the vertical motion referred to as the heave, , and the torsional degree of freedom , also known as the pitch, are shown together with the motion induced lift and moment forces, labeled and , respectively. In the following we have assumed constant mode shapes. The time-mean wind velocity is denoted .

Figure 1. Definition of the motion induced wind loads and deflections.

The equations of motion for a two degree of freedom flat plate is given in Equation (1),

where and are the modal mass and modal mass moment of inertia per unit length, and are the modal damping ratios for the heave and pitch motions, respectively. andare

Proc. of the 8th Asia-Pacific Conference on Wind Engineering Nagesh R. Iyer, Prem Krishna, S. Selvi Rajan and P. Harikrishna (eds)Copyright c 2013 APCWE-VIII. All rights reserved. Published by Research Publishing, Singapore. ISBN: 978-981-07-8011-1doi:10.3850/978-981-07-8012-8 172 400

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the cyclic eigen frequencies of the heave and pitch motions in still air, respectively. Accented dots indicated differentiation with respect to time.

(1a) (1b)

The loads are defined in Equation (2), where is the reduced non-dimensional

frequency, is deck width and is the fluid density. The non-dimensional coefficients and are referred to as the flutter derivatives or aerodynamic derivatives, ADs, as will be done in the present paper. A third degree of freedom may be introduced which corresponds to the horizontal deflection of the bridge-deck. This leads to a total of ADs, as originally introduced by Theodorsen (1934).Chowdhury and Sarkar (2003) introduced a new identification method for all ADs and highlighted some of the challenges in the experimental procedures for the three degree of freedom system. Brownjohn and Jakobsen (2001) offer a description of multiple identification methods. Multiple studies have shown that fewer ADs might be used. Bartoli and Mannini (2008) simplified the eigenvalue problem of stability and showed that, for a wide range of cases with frequency ratios not too close to unity, a total three (, and ) or even two ( and ) ADs could be used to predict the onset of flutter. In the present work we shall adopt the use of eight ADs.

(2a)

(2b)

Dyrbye and Hansen (1996) describes a solution procedure to determine the flutter wind

velocity, . Figure 2 shows the variation of the relativeflutter wind velocity, , as a function of the torsional-to-vertical frequency ratio, = , where is the natural frequency of the heave motion in still air.

Figure 2. Principal variation of the reduced critical flutter wind velocity with the

frequency ratio. In the actual design phase the critical flutter wind speed should be around 50% larger

than the characteristic 10 minute mean wind velocity at bridge-deck height (Dyrbye and Hansen (1996)). As shown in Figure 2 the flutter wind velocity is increased with the torsional-

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Proc. of the 8th Asia-Pacic Conference on Wind Engineering (APCWE-VIII)

to-vertical frequency ratio.The desired flutter wind velocities for suspension bridges are usually obtained by ensuring a sufficient torsional rigidity. Very long suspension bridges might, however, lose torsional rigidity due to the increase in span to width ratio. Large flutter wind velocities may instead be obtained by using a design in which the torsional natural frequency is deliberately lower than the vertical natural frequency as suggested by Dyrbye and Hansen(1996). In the present work this design principlewill be termed the non-flutter design principle. As noted by Dyrbye and Hansen (1996), this principle has apparently never been applied in practice.

Classical flutter of a thin airfoil using thetheorem of Theodorsen (1934), and the solution procedure in Dyrbye and Hansen (1996), does not provide an analytical solution to the critical flutter wind velocity for a torsional-to-vertical frequency ratio below approximately. The purpose of the present study is to experimentally investigate the actual section model behavior at a range of torsional-to-vertical frequency ratios spanning across this theoretical limit value of approximately . Qin et al. (2009) performed wind tunnel experiments with a streamlined twin-deck bridge section model, using two different torsional-to-vertical frequency ratios. Their investigations covered multiple separations between the twin decks, including a case without separation. It was concluded that the ADs, in the zero separation case, are not sensitive to torsional-to-vertical frequency ratios. The two ratios tested were approximately 1.2 and 2.2. The present experiments will replicate these ratios. Further, two cases with ratios below 1 will be considered in order to investigate frequency ratios on either side of the asymptotical behavior illustrated in Figure 2. The four different cases considered, are listed in Table 1.

Experimental setup and procedure

The experiments were performed in the wind tunnel at Svend Ole Hansen ApS in Copenhagen. The wind tunnel is a boundary layer tunnel of the open return flow type. The width and height of the test section area isand , respectively. A single fan in the upstream end of the tunnel creates the flow. The maximum wind speed in the wind tunnel is approximately . The exact wind speed was calculated from a Pitot tube placed upstream of the model, while taking into account the location of the model together with measured humidity, temperature and barometric pressure.

Figure 3. Experimental rig and naming convention for the force tranducers.

Figure 3 shows the experimental model-rig system.A flat plate with a height-to-width

ratio,, of spansacross the wind tunnel. Outside the wind tunnel a horizontal bar, connected to the model via a central rod, is suspended from springs at configurable positions, which allows adjustment of the torsional rigidity. See also Figure 4. Furthermore, movable

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Proc. of the 8th Asia-Pacic Conference on Wind Engineering (APCWE-VIII)

dummy masses, not to be confused with the smaller magnet mounting plates, are placed on the horizontal bar. This allowsa fine tuningof the mass moment of inertia. Hence, the spring and dummy mass configuration determines the torsional natural frequency, and the torsional-to-vertical frequency ratio.

As mentioned the critical flutter wind velocity, for a particular model, is traditionally

increased by an increase in torsional-to-vertical frequency ratio. It should be noted, that the effect of the increase depends on the way in which it is achieved. Using terms corresponding to the simplified bridge deck setup of the current experiment, the frequency ratio can be increased by either an increase of spring eccentricity or by a reduction in mass moment of inertia. The increase in critical flutter wind velocity is larger if the larger frequency ratio was obtained by increasing spring eccentricity, i.e. torsional rigidity, than when reducing the mass moment of inertia.This point should be remembered, when discussing torsional divergence wind velocity versus flutter wind velocity. (Andersen (2013)).

Figure 4.The experimental rig mounted in the wind tunnel.

The spring stiffness, , of each of the four springs was . Hence, the total vertical spring stiffness was while the torsional stiffness was calculated from , as done in Brownjohn and Jakobsen (2001), with being the spring eccentricity given in Table 1.

Table 1: Test arrangements for a single flat plate

Case 1 Case 2 Case 3 Case 4 Mass () of model-rig system 11.49 11.49 11.49 9.17 Mass moment of inertia () of model-rig system 0.51 0.33 0.18 0.18

Spring eccentricity, () 0.150 0.150 0.150 0.295 Natural Vertical frequency () 1.17 1.17 1.17 1.31 Torsional-to-vertical frequency ratio 0.71 0.88 1.19 2.10

For each of the four cases at least three free decay tests were performedin still air to

identify the properties of the individual configuration. The model was given an initial rotation and vertical displacement by usingtwo electromagnets to hold the model.The electromagnets were used to ensure a simultaneously release at both ends of the model. With an aim tocollect

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Proc. of the 8th Asia-Pacic Conference on Wind Engineering (APCWE-VIII)

data for the future determination of ADs the same setup was used to excite the model while being subjected to a rangeof wind speeds. For cases 1 and 2 the range of windspeeds tested were between and in intervals of equivalent to increment of relative wind velocity of approximately . For case 3 and 4 the experiments were conducted at the same intervals, but was stopped at wind velocities of and , respectively. The final wind speed in the individual cases corresponds to either the maximum wind speed of the tunnel, or the wind speed at which the motions of the model became unstable.To limit the possibility of random measuring errors the measurements were repeated at least three times at all wind speeds i.e. the model was reset with its initial displacements and re-measured before the windspeed was changed.

The response of the model was measured using four force transducers connected to the

model through the springs. Each force transducer was calibrated prior to measuring data in each of the four cases. The response was sampled at and translated to displacements. The heave, (), and pitch, (), degrees of freedom was determined from Eq. 3 and Eq. 4, respectively.

(3)

(4)

The model naturally has an asymmetrical mode corresponding to rotation around a line in

the stream-wise direction. The dummy masses, placed at the model ends outside of the tunnel, contributed to the lowering of the natural frequency of this asymmetrical mode, to a point where it approached the natural frequencies of the heave and pitch motions. For case 1, 2 and 3 the natural frequency of the asymmetrical mode was found to be , and was therefore larger than those of the heave and pitch motions. For case 4 however, the natural frequency of the asymmetrical mode was found to be , and it was therefore larger than the natural frequency of the heave motion () and smaller than the natural frequency of the pitch motion (. The asymmetrical mode was removed from the signals of the respective channels before calculating and . The effect of this removal is illustrated by the power spectral density shown in Figure 5, where the peak around 1.88Hz is clearly removed, without affecting the main peaks of the torsional and heave signals.

Figure 5. Spectral content of before and after the removal of the asymmetrical mode.

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Results and 'iscussion Figure 6 shows time histories for the vertical and torsional degrees of freedom in each of

the four cases at the maximum wind speed at which they were tested. Case 1 and 2 showed no signs of unstable oscillations, and the experiments was ended at the full speed of the wind tunnel. Cases 3 and 4 resulted in unstable motions, i.e. diverging oscillations, and the experiments had to be stopped. Table 2 shows the critical theoretical relative wind velocity for flutter, , the relative divergence wind velocity and the final relative wind velocity for each of the four cases.

In case 3 the initial excitation of the heave motion was quickly damped and then, together with the torsional motion, showed a clearly diverging oscillatory motion. The experiment was stopped at a relative wind velocity of due to large amplitude motions. The critical relative wind velocity for flutter was calculated to be . As both the heave and torsional motions show the diverging behavior and the theoretical value of flutter was found to be close to the velocity at which the experiment had to be stopped, we categorize the motion as being coupled, i.e. classical, flutter.

Case 4 had a frequency ratio of , which should result in a higher critical wind velocity for flutter. As shown in Figure 6, diverging oscillations could be seen for the torsional motion with amplitudes expanding well beyond the initial excitation. Further, there was a slowly diverging tendency for the heave motion. The heave motion was initially damped, but within the sample period increased to approximately of the initial excitationamplitude. Hence, case 4 showed clear signs of torsional flutter. One can consider the pure torsional motion by combining Eq.s 1b and 2b and by setting the cross terms . Torsional flutter will occur if the resulting damping, i.e. structural and aerodynamic damping combined, becomes zero. This requires to havepositive values. Using the sign convention in Dyrbye and Hansen (1996) the value of is negative for a theoretically thin airfoil, which is also true for streamlined box girder bridge decks. As noted in Dyrbye and Hansen (1996) the value of might, however,be positive for non-streamlined cross-sections which could then lead to negative torsional damping. Using the above arguments, while recalling the sharp edged bluff body geometry of the section model, the observed response in case 4 seems to suggest that the value of was positive at the time where the experiment was stopped. This should be confirmed at a later stage when aerodynamic derivatives will be identified.

Table 2: Test results for a single flat plate

Case 1 Case 2 Case 3 Case 4

The relative critical wind velocity for flutter - - 23.74 65.47 The relative divergence wind velocity 57.70 57.90 57.34 101.39 The final relative wind velocity 38.12 38.12 21.79 29.18

Case 1 and 2 had a frequency ratio below. As explained in the introduction this is unlike

traditional bridge design, where the torsional-to-vertical frequency ratio is aimed to be as high as possible, while being larger than . The time histories in Figure 6shows that the oscillations caused by the initial excitation,in these cases, were quickly damped and the model remained stable for the remainder of the sample period. The time histories are shown for the maximum relative wind velocity of the tunnel corresponding to a relative wind velocity of . Hence, no aerodynamic instabilities were observed within the range of relative velocities obtainable in the current experiments.This supports the theory of the non-flutter design principle. In the classical coupled flutter vibrations one of the main drivers are the lowering of the frequency of the torsional motion to a point where it couples with the heave motion. In the present cases the torsional frequency is already lower than the heave frequency, and is still reduced more, with an increase in relative wind velocity, than the frequency of the

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heave motion. Due to this the coupling seems unlikely to occur. A natural point of concern is of course the complete removal of torsional stiffness which would result in non-oscillatory divergence. As shown in Table 2 the relative divergence velocity for case 1 and 2 are both which is higher than the theoretical relative flutter wind velocityobtained in case 3 (frequency ratio of ) but lower than the obtained in case 4 (frequency ratio ). As a consequence one could argue that case 1 and 2 resulted in a lower theoretical design wind speed than case 4 with a frequency ratio of . However, the experiments showed that case 4 failed at a relative wind velocity of , which did therefore not result in a higher design wind speed.

The tKeoretical critical wind velocities listed in Table 2 are based on aerodynamic

derivaties corresponding to a thin airfoil. The torsional flutter occouring for in case 3 suggest that the value of should differ substantially from its theoretical value for this to occour. Based on this the values of flutterwind velocities should be recalculated using aerodynamic derivaties corresponding to the actual cross-section used.

Future work should include the use of bridge section models of existing long span bridges,

configured with a torsional-to-vertical frequency ratio below 1, to test the non flutter design principle on existing bridge section models.

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Figure 6. Time histories of the torsional () and heave () responses for each of the four cases. Here shown for the maximum wind velocity of each case, at which data was collected.

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Conclusion A simplified bridge-deck section model represented by a sharp edged 1:10 flat plate was

configured for four different torsional-to-vertical frequency ratios. Two of the cases had a torsional-to- vertical frequency ratio below 1. Within the maximum wind velocity obtainable in the tunnel these two cases did not show any sign of aerodynamic instability. Two cases with frequency ratios of 1.2 and 2.1 showed classical and torsional flutter, respectively. Based on the present experiments it can be concluded that non-flutter design principle, i.e. the use of torsional-to- vertical frequency ratios below 1, resulted in higher critical wind velocities and therefore the possibility of higher design wind speeds at least for the present cross-section design.

Acknowledgements

Svend Ole Hansen and the involved employees from Svend Ole Hansen ApS are gratefully acknowledged for their kind and helpful guidance throughout the preparation and execution of the present experiments. References Andersen, M. Styrk (2013), Non-flutter design principle Preliminary studies. Master thesis pre-

study.University of Southern Denmark.

Bartoli, G., Mannini, C. (2008), A simplified approach to bridge deck flutter, Journal of Wind Engineering and Industrial Aerodynamics, (96)2:229-256.

Brownjohn, J.M.W., Jakobsen, J.B. (2001) Strategies for aeroelastic parameter identification from bridge deck free vibration data, Journal of Wind Engineering and Industrial Aerodynamics, (89)13:1113-1136.

Chowdhury, A. G., Sarkar, P.P.(2003).A new technique for identification of eighteen flutter derivatives using a three-degree-of-freedom section model, Engineering Structures, (25)14:1763-1772.

Dyrbye, C. and Hansen, S.O. (1996), Wind loads on structures, John Wiley & Sons, New York, USA.

Simiu, E and Scanlan, R. H. (1996), Wind Effects on Structures, 3rd Edition, John Wiley & Sons, New York, USA.

Theodorsen, Th. (1935), General Theory of Aerodynamic Instability and the Mechanism of Flutter, NASA report no.496, Washington DC.

Qin X.R., Kwok K.C.S., Fok C.H. and Hitchcock P.A. (2009).Effects of frequency ratio on bridge aerodynamics determined by free-decay sectional model tests. Wind and Structures, 12(5):413-424.

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