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Aerodynamic stability of non-symmetrically erected suspension bridge girders
Journal of Wind Engineering and Industrial Aerodynamics 80 (1999) 85—104
Aerodynamic stability of non-symmetrically erected suspension bridge girders H. Tanaka , N.J. Gimsing Department of Civil Engineering, University of Ottawa, Ottawa, Canada K1N 6N5 Department of Structural Engineering and Materials, Technical University of Denmark, DK-2800 Lyngby, Denmark Received 6 January 1997; accepted 2 July 1998
Abstract An aeroelastic model of a suspension bridge at its deck erection stages was wind tunnel tested. With a shallow streamlined box as its deck cross-section, the flutter-type instability was the main concern. An interesting observation was that the bridge was found to be more stable when the suspended deck was extended in a non-symmetric manner as opposed to the traditional symmetric erection schedule. It is possible to use non-symmetric deck configurations as a stabilizing measure during the deck erection. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Wind; Suspension bridges; Deck erection; Stability; Wind tunnel tests
Notation B H I J K ¸ M M P º a d f m r s
deck width horizontal component of cable tension turbulence intensity mass moment of inertia per unit deck length reduced frequency defined by 2pfB/º span length or aerodynamic lift force per unit deck length mass pitching moment point load mean wind speed at the deck level loading point ("k¸) cable sag frequency mass per unit deck length radius of gyration sag ratio ("d/¸)
0167-6105/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 1 9 7 - 4
86
u, w w x b c f g j k, l m o
A*, H* H H ¸ , ¸ S U
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fluctuating velocity components in along-wind and vertical directions, respectively weight per unit length coordinate along the structure dimensionless parameter defined by Eq. (A.2) dimensionless parameter defined by 1#(H /H ) N U structural damping given as a fraction of critical cable deflection dimensionless parameter defined by P/(w¸) dimensionless mass parameters dimensionless coordinate defined by x/¸ air density made shape function aerodynamic derivatives ( j"1, 2, 3, 4) length scales of turbulence
Subscripts C D D F P S T V e p w
cables deck part of the deck to be considered flutter conditions pylons the whole structure torsion vertical bending “effective” values live load dead load
1. Introduction The maximum danger of flutter-type aerodynamic instability of suspension bridges occurs during the early erection stage of the deck because of the significant reduction of overall torsional stiffness in particular. It has been an important engineering issue which has attracted considerable attention during the last 30 years. A comprehensive overview of the subject has been presented by Brancaleoni [1] discussing various factors to be considered such as evolution of frequencies, deck stiffness, structural as well as artificial dampings, the influence of finite deck lengths and provision of eccentric mass. The analytical aspect of the mass eccentricity across the bridge deck, which was a measure adopted for erection of the Humber bridge, has been elaborated by Larsen [2]. The influence of vibration mode shapes has been discussed, for example, by Svensson [3]. The present paper focusses its attention only on self-excited instability of the suspension bridge deck during the deck erection, and other dynamic aspects such as buffeting or vortex excitation are not included in its scope. The experimental findings discussed in the present paper are from the aeroelastic wind tunnel tests of the Ho¨ga Kusten bridge carried out at the Danish Maritime Institute [4,5].
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2. Description of the bridge The Ho¨ga Kusten bridge is a 1800 m suspension bridge with a main span of 1210 m, with a streamlined box girder and is currently under construction in Sweden. An elevation of the bridge and its deck cross-section are shown in Fig. 1. The bridge deck is prefabricated in 40 m long sections, transported to the construction site and lifted one by one. The proposed sequence of hoisting the segments for the centre span is to start from the midspan and extend alternatively towards both pylons, one direction at a time. Because of this erection schedule, the actual deck configurations are expected often to be non-symmetric as opposed to the conventional symmetric style [2]. Initially, the box segments will be connected with temporary hinges which allows the full transfer of torsional and lateral stiffnesses whereas the vertical bending is left free. They will not be welded together until all boxes are in position. The side span boxes are already erected before the centre span erection starts, first supported on an intermediate scaffolding. The side spans will be lifted up when the centre span is sagged down with about two-third of its span loaded. Details of the construction procedure is given elsewhere [6].
3. Consideration of stiffness provided by cables with partially erected deck In the case of a long span suspension bridge such as this, the vertical bending stiffness is provided mostly by the main cables. Particularly during the deck erection,
Fig. 1. Ho¨ga Kusten bridge.
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with the initial joining of the box segments by temporary hinges, the hoisted deck segments add weight but hardly influence on the vertical bending stiffness. In torsional oscillations, however, where the two main cables oscillate out-of-phase, the torsional stiffness of the bridge deck will give a substantial restraint against mutual twisting of the cables, even during the deck erection, in the region where the girder is in place. The difference in the way the partially erected bridge deck affects the behaviour of the vertical cable planes for vertical and torsional oscillations is indicated in Fig. 2. The actual structural system is given in (a) and the cable model to be used for determining the characteristics of the vertical modes is shown in (b), whereas (c) is the cable model for the torsional modes. In stages where the deck erection is confined to the central half of the main span and the deck comprises a wide box girder, it might be assumed that under torsional oscillations there is no change of curvature of the main cable in the region where the deck is in place and also that this part will move in translation as a rigid body. In the initial erection phase with only a short length of the deck added, the effect of the restraint offered by the torsional stiffness of the deck will be modest, but with increased deck length this effect will be more and more pronounced.
Fig. 2. Modelling of cable-deck system.
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In a comparison of symmetric and non-symmetric suspension of the partially erected deck, it can be pointed out that the torsional restraint offered by the two main cables will increase in the non-symmetric cases, as the shortened free main cables on one side add more stiffness than that lost by the increased length of the free main cables on the other side. A simple explanation of this effect is given briefly in Appendix A. As an extreme case, the eccentricity of the erected deck could be so large that one end of the suspended deck would reach the end of the span. Torsional oscillations for this case would be characterized by very high frequency values due to the efficient restraint offered by the torsional rigidity of the box. An optimum torsional stiffness of the deck in the temporary erection stages has in some cases been achieved by erecting the deck from the pylons towards midspan — in most cases by moving out symmetrically from both pylons [7]. Such a procedure, however, has its own shortcoming; i.e., larger bending stresses in the main cables result from the requirement of tightening the cable bands adjacent to the pylons right from the beginning. As the deck erection proceeds, substantial angular changes take place in the main cables since the movements of the wire inside the cables are restrained by the clamping force of the cable bands. If deck erection is performed from midspan, the outer cable bands can remain untightened or modestly tightened until the last girder segments adjacent to the pylons are lifted into place.
4. Experimental observations 4.1. Description of the aeroelastic model test A full aeroelastic model test of the Ho¨ga Kusten bridge at its deck erection stages was carried out in the Martin Jensen wind tunnel (13.6 m wide;20 m long) of the Danish Maritime Institute. Both, the bridge model and the simulated natural wind were designed to a linear scale of 1 : 150 and other dimensions were decided based on the Froude number scaling [4]. In order to simulate the deck stiffness at each erection stage properly, the stiffening spine of the bridge deck was made of a number of short segments connected by hinges. The model was exposed to a boundary layer wind flow which was designed to simulate natural wind conditions at the bridge site. The wind induced response was measured by detecting displacements, accelerations and bending moments at various locations of the deck. The tests were carried out in two phases. The first series was a contract research and particular construction stages shown in Fig. 2 were faithfully modelled as specified by the client [4]. The second series was an extension of the first phase, carried out as a research project to investigate particularly the effect of non-symmetric deck configurations on the flutter stability limit. The structural configuration in this phase was changed in various ways, including different deck lengths and cases with the same deck length but at different locations, including both symmetric and non-symmetric cases with various degrees of eccentricity.
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A deck configuration can be expressed by two dimensionless parameters. The degree of completion is given by ¸ /¸, where ¸ is the length of the erected deck and ¸ is the centre span length corresponding to 1210 m in full scale. Another parameter is defined by D/¸ , where D is the distance between the centre of the erected deck and the midspan, showing the degree of eccentricity in deck distribution. A total of 27 cases in the range 0.033)¸ /¸)1.00 were wind tunnel tested, of which 6 were on the symmetric deck configurations. 4.2. Test results The stability limits found for various deck configurations are summarized in Figs. 3 and 4. Fig. 3 shows the change of critical flutter speed with the percentage of deck completion. Symmetric and non-symmetric deck configurations are distinguished by black and white marks, but the degree of eccentricity is not shown in this figure. Fig. 4, on the other hand, shows the effect of D/¸ . However, only the cases with similar ¸ /¸ ratio can be compared in this manner. The following observations can be made on them: (1) The stability limit is generally lower through the erection period compared to what is expected for the bridge in service and increases with the percentage of deck completion.
Fig. 3. Development of critical stability limit.
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Fig. 4. Variation of stability limit due to deck eccentricity.
(2) The critical flutter speed becomes the lowest at a relatively early stage when 10—15% of the deck is suspended. (3) The flutter limit is very high when the deck is very short at the earliest erection stages. (4) The lowest stability limit for a given length of erected main span deck is observed when the deck is symmetric about the midspan. The first two observations were more or less expected. The third point can be easily understood since it should be caused by the effective mass contribution as elaborated in a later section, and also possibly by aerodynamic end effects. The most interesting point is the last listed above. It is a new finding and has intrigued the authors very much raising the question of why it happens. The test results clearly indicate that, at least for this particular bridge, it is possible to choose the erection procedure in a way so that a relatively high stability limit can be maintained all through the procedure [5]. An interesting question, of course, is if asymmetric deck installation can generally be considered as a measure to stabilize a bridge during deck erection.
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5. Discussion on the test results 5.1. General trend: frequency effect It has been widely acknowledged that the stability limit is lower through the erection period than after completion [1,2]. It can be generally attributed to the lack of overall torsional stiffness when the deck is incomplete as discussed in the previous section. The bending stiffness, on the other hand, is less sensitive to the degree of deck erection and this results in an increase of the frequency ratio f /f . 2 4 When the deck cross-section is a streamlined shallow box, it is known that the use of the Selberg formula [7]
º mr f $ "3.7 1! 4 Bf oB f 2 2
(1)
would give the flutter speed with good accuracy, provided that f /f is not close to 4 2 unity. In Eq. (1), B is the width of the deck, r is the radius of gyration of the deck, m is the mass per unit length of the deck, o is the air density and º is the the flutter speed. $ The significant reduction of stability limit which has been observed in the first half of the erection period can partly be explained by the change in the frequency ratio. There have been a few reports showing a second, less significant reduction of the stability limit when about of the deck is erected [1] and it may be so for this bridge as well, though the second-half was not fully examined in this study. It may be more accurate to state that the stability limit is generally not expected to rise much once half of the span is already in place. The lowest critical speed for this bridge was found with 10—15% of the deck erected. The tendency is consistent with what has been found for other bridges [1,2,8] though the minimum stability limit has sometimes been found a little later at 20—30% completion. 5.2. Stability of very short decks Frequency Ratio: The significantly high stability limit for very short deck lengths may be caused by various reasons, but is generally considered to be due to the extreme reduction of the frequency ratio which approaches unity. If the suspended deck is very short, its vertical and torsional vibrations are essentially in-phase and out-of-phase vibrations of the cables, and their frequencies are very close to each other. The flutter speed is known to increase drastically when the frequency ratio approaches unity or very close to it, and Eq. (1) is not applicable for this case [8]. Effective Mass: The increase of flutter speed is also caused by the small aerodynamic excitation relative to the large inertial force. The aerodynamic excitation is supplied only through a short bridge deck but the vibration inertia is governed not only by the deck itself but also by the cables and pylons. The concept of “effective mass” can be introduced to give a clearer understanding regarding this point. This is the same concept as the calculation of mass required for sectional model wind tunnel tests.
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Consider the distributed mass of the structure m(x). When the vibration mode is expressed by (x), the generalized mass for this particular mode of vibration for the structure is given by
M " m(x) (x) dx, 1 1 where the integration is to be done not only over the bridge deck but over the whole bridge, including the cables and pylons. Eq. (1) considers a two-dimensional slice of the bridge just like the idea of the conventional sectional model wind tunnel tests, where constant mass per unit length (m) is assumed over the deck length and other bridge components are not considered, with the understanding that the aerodynamic forces on other components are not contributing to the instability. Hence the generalized mass for this idealized system is given by
M " m (x) dx, "Y "Y "Y where m is the required effective mass. The range of integration D refers to the part of the deck represented or the deck length considered. The mode shape is defined by taking the maximum deck displacement as unity. Since two generalized masses, M and M , must be identical, the required magnitude of the effective mass m is 1 "Y decided as follows: m (x) dx# m (x) dx# m (x) dx ! ! ! . . . m"" " " , (2) (x) dx "Y "Y where subscripts D, C and P refer to the deck, cables and pylons. The effective mass m should be the mass parameter m applied in Eq. (1). The effective mass moment of inertia J can be defined in a similar way. The total mass contribution in relation to the amount of energy supplied by the aerodynamic excitation for each mode can be thus expressed by a simple parameter m r /(oB), where r "(J /m ) . The change of this parameter, together with the change of eigen frequencies of the modes which are participating in the flutter instability for the proposed erection procedure of the Ho¨ga Kusten bridge are shown in Fig. 5. The flutter speed is expected to increase in proportion to the square root of this parameter as it appears in Eq. (1). 5.3. Flutter analysis Together with the experimentally obtained data, Fig. 6 shows the calculated flutter wind speeds both from 2D and 3D analyses. The 2D analysis is from Eq. (1). The change of mass, mass moment of inertia and frequency ratio are taken into account in this expression but it is not applicable when the two frequencies are very close to each
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Fig. 5. Development of frequencies and effective mass.
other. The 3D analysis is based on measured aerodynamic derivatives for the Ho¨ga Kusten bridge [4] and the vibration mode shapes given by the dynamic analysis [9]. A flutter condition was not reached for stages 1 and 7 (¸ /¸"0.05 and 0.68) in the 3D analysis.
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Fig. 6. Comparison of critical flutter speeds.
The aerodynamic derivatives were measured using a conventional sectional model for which the aerodynamic interaction is considered to be two-dimensional, or the aspect ratio (span/width) of the bridge is assumed to be infinity. They were also measured in a smooth air stream without any simulation of wind turbulence. However, at very early stages of erection, the assumption of two-dimensionality does not hold, since the aspect ratio is actually very small. For example, the aspect ratio is only 2.75 when ¸ /¸ is 5%. The aerodynamic derivatives for this section were measured separately, using a short span model with open edges, so that the three-dimensional edge effect is involved in the evaluation of the forces. It was found that the characteristics of the aerodynamic derivatives measured in this way were similar to the two-dimensional results, but their magnitude, particularly of the pitching moment components, was significantly reduced due to edge effects. As a result, the flutter speed becomes 5—10% higher. The measurement of aerodynamic derivatives on shorter models and the evaluation of flutter speeds are summarized in Appendix B. The predicted flutter speeds clearly show the tendency of the change in stability limit. The disagreement with the experimental results may be attributed to the effects of turbulence on aerodynamic derivatives which was ignored in their measurement and possible inaccuracy in calculated mode shapes which are different from the original eigen vibration modes once the flutter condition is reached.
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5.4. Effect of non-symmetry in deck configuration The effect of nonsymmetry in deck configuration is more complicated. The comparison of four different deck configurations shown in Fig. 7 and Table 1 is of some help. All four cases, A, B, C and D, have equal deck length, ¸ /¸"0.31, and the cables are rigidly fixed to the deck at the midspan, but the deck eccentricity is D/¸ "0, 0.11, 0.21 and 0.32, respectively. Case A has a symmetric deck and exhibits the lowest frequency ratio and the lowest structural damping. Both bending and torsion are in the first symmetric modes. The eccentricity of deck distribution is increased in the sequence of B, C and D. Note that both frequency ratio and damping are found to increase in this order, contributing to raising the stability limit. For cases C and D, the second torsional mode participated in the flutter whereas the first torsion was coupled in cases A and B. From a review of the mode shapes, it is apparent that the second torsional mode is asymmetric for the cables, but is found to be practically symmetric for the deck, particularly when the deck distribution is eccentric, and the effective mass parameter becomes less for these
Fig. 7. Case of 31% deck with various eccentricity.
Table 1 Dynamic characteristics of 31% deck models Case
D/¸
f (Hz) 4
f (%) 4
f (Hz) 2
f (%) 2
º (m/s) $
f (Hz) $
A B C D
0.00 0.11 0.21 0.32
0.150 0.149 0.146 0.145
0.5 0.9 0.7 0.8
0.204 0.208 0.260 0.237
0.6 0.9 1.5 1.6
57 58 64 65
0.201 0.207 0.221 0.222
Note: f , f — natural frequencies in bending and torsion. 4 2 f , f — structural damping as fraction of critical. 4 2 º , f — flutter speed and flutter frequency. $ $
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mode combinations. When actual instability occurred, the motion was heavily coupled with the lateral sway, also, though the prevailing frequency was close to the torsional frequency. This clearly indicates the fact that the flutter vibration mode cannot be simply described by the eigen vibration modes. The increase of structural damping for complicated mode shapes may be associated with the enhanced distortion of cables and other structural components in real structures. Davenport and Larose [10] have given an interesting explanation of this mechanism for suspension bridge cables. However, how much the structural damping could influence the flutter speed is a different issue. Another interesting case was with 51% of the deck erected. The non-symmetric case with the deck eccentricity of D/¸ "0.13 actually started a fairly violent coupled vibration in bending and torsion at the middle of the span with almost the same onset speed as in the case of the symmetric deck. However, the vibration was observed only locally near the midspan and did not develop to include the entire span. At the further end of the deck, only the lateral sway motion was seen as predominant. It may have been caused by the fact that, as mentioned earlier, the torsional stiffness provided by the cable system is expected to be higher near the pylons compared with the centre of the span. This phenomenon continued at even higher wind speeds until suddenly the divergence-type of destruction occurred. It was an instantaneous catastrophe, and not the gradual increase of vibration amplitude. The deck non-symmetry for this case seems to have increased the post-flutter endurance capacity of the structure.
6. Concluding remarks A series of wind tunnel tests of a suspension bridge with partially erected deck of various lengths was carried out. The flutter stability limit was found to increase when the deck had non-symmetric configurations as opposed to the traditional symmetric distribution. The reason of this phenomenon seems to be very complicated: a combination of vibration modes, frequencies and damping which are all influenced by the deck configurations. Only a three-dimensional modelling of the structure can reveal such behaviour properly. The study also suggested an interesting approach to improve the aerodynamic stability of a bridge during its deck erection. It is possible to choose non-symmetric deck configurations deliberately to maintain a higher stability limit.
Acknowledgements The wind tunnel tests were initially sponsored by the Scandinavian Bridge Joint Venture and its extension was supported by the Danish Maritime Institute (DMI). A. Damsgaard, F.M. Livesey, G.L. Larose and other DMI personnel were involved in the project. The second part of the experiment was largely conducted by P. Reina of
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Politechnico di Milano, whose study at DTU/DMI was made possible through the ERASMUS program, worked out by Dr. C. Dyrbye of the Technical University of Denmark (DTU). Useful discussion was made by Messrs. N. Franck, G. Jensen and B.S. Madsen of Monberg and Thorsen A/S, Copenhagen. The authors extend their appreciation to all of them.
Appendix A. Deflection of a cable under point load Consider a single span cable shown in Fig. 8. It is well known that the deflected shape of the cable is expressed by w¸ m(1!m), y(m)" 2H U
(A.1)
where ¸ is the span, w the weight of the cable per unit length, H the horizontal U component of the cable tension and m"x/¸. In addition to this, if a single concentrated load P is applied at distance a"k¸ from the left support A, the deflection of the cable at the loading point (m"k) is given by w¸ g(m)" bk(1!k), 2H U
b"(1#2j)/c!1,
(A.2)
where j"P/(w¸), c"1#H /H and H the horizontal component of the cable . U . tension due to P. An analytical expression for c is as follows: c(j,k)"[1#12j(1#j) (k!k)] .
Fig. 8. Deflection of a cable under point load.
(A.3)
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When the parameter j is less than 0.5 (say), the following expression can be used with good accuracy: b"0.075(j) /k.
(A.4)
Substituting Eq. (A.3) in Eq. (A.2) and also considering that H "w¸/(8s), s"d/¸ (sag ratio) U
(A.5)
the deflection due to P is given by 0.3Ps (1!k). g(k)" (j) w
(A.6)
Hence the stiffness of the cable in the case of an eccentric point loading is inversely proportional to the length of an increased half of the cable, ¸(1!k). The deck suspended with eccentric distribution would thus have less deflection for the same mass. In the case of torsional oscillations of a partially suspended deck with a certain length, the girder extended closer to one of the pylons is subjected to much higher cable stiffness and, since the deck itself has a large torsional stiffness, the overall stiffness is considered to become quite high.
Appendix B. Aerodynamic derivatives with end effects B.1. Aerodynamic derivatives The aerodynamic derivatives of the Ho¨ga Kusten sectional model were measured with its aspect ratio changed in four steps [11]. The measurement was carried out in the boundary layer wind tunnel of the Danish Maritime Institute by applying the systems identification technique which was developed earlier at DMI [12]. Experimental results are expressed in the dimensionless coefficients defined by Scanlan as follows:
oº hQ BaR h ¸" 2B KH* #KH* #KH*a#KH* , º º B 2
oº hQ BaR h M" 2B KA* #KA* #KA*a#KA* , º º B 2
(B.1)
where ¸ and M are the lift force and pitching moment per unit length of the deck, º the mean wind speed, B and ¼ along-wind and across-wind length of the deck and H*, H*,2, A* are the derivatives given as functions of reduced frequency K"2pfB/º in which f is the frequency (Hz). The aspect ratio of the model (¼/B) was 2, 3, 4 or 5. ¼/B"5 corresponded to the wall to wall or 2D case and hence could be
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Fig. 9. Aerodynamic derivatives, ¼/B"2.
regarded ideally as the same as ¼/BPR. The measurements were carried out in the Reynolds number range of 2;10—4;10. The wind flow was smooth with a longitudinal turbulence intensity of less than 1%. Test results are summarized in Figs. 9—12.
B.2. Stability limits The flutter stability limit for each ¼/B ratio was examined by free vibration tests in turbulent flow. It was also calculated using the measured aerodynamic derivatives above. The results are shown in Table 2. The stability limits in Table 2 are listed in terms of reduced velocity º "º /(Bf ). 0 $ 2 All through the tests, the frequency ratio of torsional to bending motion of the system was kept constant, at f /f "1.55. Also the dimensionless mass and mass moment 2 4
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Fig. 10. Aerodynamic derivatives, ¼/B"3.
parameters, k"m/(oB) and l"J/(oB), were kept constant with values of 17.5 and 2.06, respectively, where m and J are the mass and mass moment of inertia of the model per unit deck length. Hence the results shown in Table 2 can be considered as the effects of having the open ends of the model or the effects of its “threedimensionality”. The calculated flutter stability limits are generally found to be 4—8% lower than the directly measured results. This difference was perhaps caused by two reasons: one is the existence of flow turbulence for the case of direct measurement and another is possible errors in the measurement of the aerodynamic derivatives. The measurement of flutter speed in the dynamic test has less chance of being contaminated by experimental errors. The flow turbulence in the dynamic tests can be approximately characterized by the intensities and scales of I "6.3%, S
I "5.7%, U
V¸ "0.40 m, S
V¸ "0.17 m, U
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Fig. 11. Aerodynamic derivatives, ¼/B"4.
whereas the smooth flow had the turbulence intensity of less than 1%. The flow conditions at the deck level in the Martin Jensen wind tunnel, as a comparison, were I "11.0%, S
I "8.0%, U
V¸ "1.06 m, S
V¸ "0.11 m. U
As it is obvious from these results, the shorter the deck is, the more end effects become predominant. Or it can be considered as a deviation from the conventional concept of strip theory. The aspect ratio ¼/B"2 corresponds to a full-scale deck length of 44 m or ¸ /¸"0.036 for the case of the full bridge model, which is similar to what was tested as the shortest deck. For this case, the expected end effects on the measured flutter speed is of the order of 6% judged from these results.
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Fig. 12. Aerodynamic derivatives, ¼/B"5 (R).
Table 2 Stability limits versus aspect ratio Aspect ratio (¼/B)
2.0
3.0
4.0
5.0 (R)
Measurement (º ) 0 Calculation (º ) 0
7.3 7.0
7.2 6.9
7.1 6.8
6.9 6.3
References [1] F. Brancaleoni, The construction phase and its aerodynamic issues, Proc. Int. Symp. on Aerodynamics of Large Bridges, Copenhagen, February 1992, pp. 147—158. [2] A. Larsen, Prediction of aeroelastic stability of suspension bridges during erection, Proc. 9th ICWE, vol. 2, New Delhi, January 1995, pp. 917—927.
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[3] H.S. Svensson, I. Kovacs, Examples of analytical aerodynamic investigations of long-span bridges, Proc. Int. Symp. on Aerodynamics of Large Bridges, Copenhagen, February 1992, pp. 171—182. [4] F.M. Livesey, Wind tunnel studies of the Ho¨ga Kusten Bridge during construction, Part II: Aeroelastic model tests, DMI Report 94124, Danish Maritime Institute, June 1995. [5] H. Tanaka, A. Damsgaard, P. Reina, N. Franck, B.S. Madsen, Aerodynamic stability of a suspension bridge with partially constructed bridge decks, Presented at 15th IABSE Congress, Copenhagen, June 1996. [6] C. Pedersen, Construction of superstructure of the Ho¨ga Kusten Bridge — suspension bridge in Sweden with a 1210 m main span, Proc. Conf. Bridge into the 21st Century, Hong Kong, October 1995, pp. 243—250. [7] N.J. Gimsing, Cable Supported Bridges: Concept and Design, Wiley, New York, 1983,1996. [8] I.P. Smith, The aeroelastic stability of the Severn suspension bridge, NPL Aero Report 1105, May 1964. [9] A. Damsgaard, Ho¨ga Kusten Bridge dynamic structural analysis, DMI Report 94115, Danish Maritime Institute, April 1995. [10] A.G. Davenport, G.L. Larose, The structural damping of long span bridges: an interpretation of observations, Proc. Canada—Japan Workshop on Bridge Aerodynamics, Ottawa, September 1989, pp. 111—118. [11] P. Reina, The aerodynamic stability of a suspension bridge during erection, M.Sc. thesis, Politecnico di Milano, April 1996. [12] N.K. Poulsen, System identification in determination of the flutter coefficients, Status Report, Skibsteknisk Laboratorium, 1990.
Aerodynamic stability of non-symmetrically erected suspension bridge girders H. Tanaka , N.J. Gimsing Department of Civil Engineering, University of Ottawa, Ottawa, Canada K1N 6N5 Department of Structural Engineering and Materials, Technical University of Denmark, DK-2800 Lyngby, Denmark Received 6 January 1997; accepted 2 July 1998
Abstract An aeroelastic model of a suspension bridge at its deck erection stages was wind tunnel tested. With a shallow streamlined box as its deck cross-section, the flutter-type instability was the main concern. An interesting observation was that the bridge was found to be more stable when the suspended deck was extended in a non-symmetric manner as opposed to the traditional symmetric erection schedule. It is possible to use non-symmetric deck configurations as a stabilizing measure during the deck erection. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Wind; Suspension bridges; Deck erection; Stability; Wind tunnel tests
Notation B H I J K ¸ M M P º a d f m r s
deck width horizontal component of cable tension turbulence intensity mass moment of inertia per unit deck length reduced frequency defined by 2pfB/º span length or aerodynamic lift force per unit deck length mass pitching moment point load mean wind speed at the deck level loading point ("k¸) cable sag frequency mass per unit deck length radius of gyration sag ratio ("d/¸)
0167-6105/99/$ — see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 1 9 7 - 4
86
u, w w x b c f g j k, l m o
A*, H* H H ¸ , ¸ S U
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fluctuating velocity components in along-wind and vertical directions, respectively weight per unit length coordinate along the structure dimensionless parameter defined by Eq. (A.2) dimensionless parameter defined by 1#(H /H ) N U structural damping given as a fraction of critical cable deflection dimensionless parameter defined by P/(w¸) dimensionless mass parameters dimensionless coordinate defined by x/¸ air density made shape function aerodynamic derivatives ( j"1, 2, 3, 4) length scales of turbulence
Subscripts C D D F P S T V e p w
cables deck part of the deck to be considered flutter conditions pylons the whole structure torsion vertical bending “effective” values live load dead load
1. Introduction The maximum danger of flutter-type aerodynamic instability of suspension bridges occurs during the early erection stage of the deck because of the significant reduction of overall torsional stiffness in particular. It has been an important engineering issue which has attracted considerable attention during the last 30 years. A comprehensive overview of the subject has been presented by Brancaleoni [1] discussing various factors to be considered such as evolution of frequencies, deck stiffness, structural as well as artificial dampings, the influence of finite deck lengths and provision of eccentric mass. The analytical aspect of the mass eccentricity across the bridge deck, which was a measure adopted for erection of the Humber bridge, has been elaborated by Larsen [2]. The influence of vibration mode shapes has been discussed, for example, by Svensson [3]. The present paper focusses its attention only on self-excited instability of the suspension bridge deck during the deck erection, and other dynamic aspects such as buffeting or vortex excitation are not included in its scope. The experimental findings discussed in the present paper are from the aeroelastic wind tunnel tests of the Ho¨ga Kusten bridge carried out at the Danish Maritime Institute [4,5].
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2. Description of the bridge The Ho¨ga Kusten bridge is a 1800 m suspension bridge with a main span of 1210 m, with a streamlined box girder and is currently under construction in Sweden. An elevation of the bridge and its deck cross-section are shown in Fig. 1. The bridge deck is prefabricated in 40 m long sections, transported to the construction site and lifted one by one. The proposed sequence of hoisting the segments for the centre span is to start from the midspan and extend alternatively towards both pylons, one direction at a time. Because of this erection schedule, the actual deck configurations are expected often to be non-symmetric as opposed to the conventional symmetric style [2]. Initially, the box segments will be connected with temporary hinges which allows the full transfer of torsional and lateral stiffnesses whereas the vertical bending is left free. They will not be welded together until all boxes are in position. The side span boxes are already erected before the centre span erection starts, first supported on an intermediate scaffolding. The side spans will be lifted up when the centre span is sagged down with about two-third of its span loaded. Details of the construction procedure is given elsewhere [6].
3. Consideration of stiffness provided by cables with partially erected deck In the case of a long span suspension bridge such as this, the vertical bending stiffness is provided mostly by the main cables. Particularly during the deck erection,
Fig. 1. Ho¨ga Kusten bridge.
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with the initial joining of the box segments by temporary hinges, the hoisted deck segments add weight but hardly influence on the vertical bending stiffness. In torsional oscillations, however, where the two main cables oscillate out-of-phase, the torsional stiffness of the bridge deck will give a substantial restraint against mutual twisting of the cables, even during the deck erection, in the region where the girder is in place. The difference in the way the partially erected bridge deck affects the behaviour of the vertical cable planes for vertical and torsional oscillations is indicated in Fig. 2. The actual structural system is given in (a) and the cable model to be used for determining the characteristics of the vertical modes is shown in (b), whereas (c) is the cable model for the torsional modes. In stages where the deck erection is confined to the central half of the main span and the deck comprises a wide box girder, it might be assumed that under torsional oscillations there is no change of curvature of the main cable in the region where the deck is in place and also that this part will move in translation as a rigid body. In the initial erection phase with only a short length of the deck added, the effect of the restraint offered by the torsional stiffness of the deck will be modest, but with increased deck length this effect will be more and more pronounced.
Fig. 2. Modelling of cable-deck system.
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In a comparison of symmetric and non-symmetric suspension of the partially erected deck, it can be pointed out that the torsional restraint offered by the two main cables will increase in the non-symmetric cases, as the shortened free main cables on one side add more stiffness than that lost by the increased length of the free main cables on the other side. A simple explanation of this effect is given briefly in Appendix A. As an extreme case, the eccentricity of the erected deck could be so large that one end of the suspended deck would reach the end of the span. Torsional oscillations for this case would be characterized by very high frequency values due to the efficient restraint offered by the torsional rigidity of the box. An optimum torsional stiffness of the deck in the temporary erection stages has in some cases been achieved by erecting the deck from the pylons towards midspan — in most cases by moving out symmetrically from both pylons [7]. Such a procedure, however, has its own shortcoming; i.e., larger bending stresses in the main cables result from the requirement of tightening the cable bands adjacent to the pylons right from the beginning. As the deck erection proceeds, substantial angular changes take place in the main cables since the movements of the wire inside the cables are restrained by the clamping force of the cable bands. If deck erection is performed from midspan, the outer cable bands can remain untightened or modestly tightened until the last girder segments adjacent to the pylons are lifted into place.
4. Experimental observations 4.1. Description of the aeroelastic model test A full aeroelastic model test of the Ho¨ga Kusten bridge at its deck erection stages was carried out in the Martin Jensen wind tunnel (13.6 m wide;20 m long) of the Danish Maritime Institute. Both, the bridge model and the simulated natural wind were designed to a linear scale of 1 : 150 and other dimensions were decided based on the Froude number scaling [4]. In order to simulate the deck stiffness at each erection stage properly, the stiffening spine of the bridge deck was made of a number of short segments connected by hinges. The model was exposed to a boundary layer wind flow which was designed to simulate natural wind conditions at the bridge site. The wind induced response was measured by detecting displacements, accelerations and bending moments at various locations of the deck. The tests were carried out in two phases. The first series was a contract research and particular construction stages shown in Fig. 2 were faithfully modelled as specified by the client [4]. The second series was an extension of the first phase, carried out as a research project to investigate particularly the effect of non-symmetric deck configurations on the flutter stability limit. The structural configuration in this phase was changed in various ways, including different deck lengths and cases with the same deck length but at different locations, including both symmetric and non-symmetric cases with various degrees of eccentricity.
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A deck configuration can be expressed by two dimensionless parameters. The degree of completion is given by ¸ /¸, where ¸ is the length of the erected deck and ¸ is the centre span length corresponding to 1210 m in full scale. Another parameter is defined by D/¸ , where D is the distance between the centre of the erected deck and the midspan, showing the degree of eccentricity in deck distribution. A total of 27 cases in the range 0.033)¸ /¸)1.00 were wind tunnel tested, of which 6 were on the symmetric deck configurations. 4.2. Test results The stability limits found for various deck configurations are summarized in Figs. 3 and 4. Fig. 3 shows the change of critical flutter speed with the percentage of deck completion. Symmetric and non-symmetric deck configurations are distinguished by black and white marks, but the degree of eccentricity is not shown in this figure. Fig. 4, on the other hand, shows the effect of D/¸ . However, only the cases with similar ¸ /¸ ratio can be compared in this manner. The following observations can be made on them: (1) The stability limit is generally lower through the erection period compared to what is expected for the bridge in service and increases with the percentage of deck completion.
Fig. 3. Development of critical stability limit.
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Fig. 4. Variation of stability limit due to deck eccentricity.
(2) The critical flutter speed becomes the lowest at a relatively early stage when 10—15% of the deck is suspended. (3) The flutter limit is very high when the deck is very short at the earliest erection stages. (4) The lowest stability limit for a given length of erected main span deck is observed when the deck is symmetric about the midspan. The first two observations were more or less expected. The third point can be easily understood since it should be caused by the effective mass contribution as elaborated in a later section, and also possibly by aerodynamic end effects. The most interesting point is the last listed above. It is a new finding and has intrigued the authors very much raising the question of why it happens. The test results clearly indicate that, at least for this particular bridge, it is possible to choose the erection procedure in a way so that a relatively high stability limit can be maintained all through the procedure [5]. An interesting question, of course, is if asymmetric deck installation can generally be considered as a measure to stabilize a bridge during deck erection.
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5. Discussion on the test results 5.1. General trend: frequency effect It has been widely acknowledged that the stability limit is lower through the erection period than after completion [1,2]. It can be generally attributed to the lack of overall torsional stiffness when the deck is incomplete as discussed in the previous section. The bending stiffness, on the other hand, is less sensitive to the degree of deck erection and this results in an increase of the frequency ratio f /f . 2 4 When the deck cross-section is a streamlined shallow box, it is known that the use of the Selberg formula [7]
º mr f $ "3.7 1! 4 Bf oB f 2 2
(1)
would give the flutter speed with good accuracy, provided that f /f is not close to 4 2 unity. In Eq. (1), B is the width of the deck, r is the radius of gyration of the deck, m is the mass per unit length of the deck, o is the air density and º is the the flutter speed. $ The significant reduction of stability limit which has been observed in the first half of the erection period can partly be explained by the change in the frequency ratio. There have been a few reports showing a second, less significant reduction of the stability limit when about of the deck is erected [1] and it may be so for this bridge as well, though the second-half was not fully examined in this study. It may be more accurate to state that the stability limit is generally not expected to rise much once half of the span is already in place. The lowest critical speed for this bridge was found with 10—15% of the deck erected. The tendency is consistent with what has been found for other bridges [1,2,8] though the minimum stability limit has sometimes been found a little later at 20—30% completion. 5.2. Stability of very short decks Frequency Ratio: The significantly high stability limit for very short deck lengths may be caused by various reasons, but is generally considered to be due to the extreme reduction of the frequency ratio which approaches unity. If the suspended deck is very short, its vertical and torsional vibrations are essentially in-phase and out-of-phase vibrations of the cables, and their frequencies are very close to each other. The flutter speed is known to increase drastically when the frequency ratio approaches unity or very close to it, and Eq. (1) is not applicable for this case [8]. Effective Mass: The increase of flutter speed is also caused by the small aerodynamic excitation relative to the large inertial force. The aerodynamic excitation is supplied only through a short bridge deck but the vibration inertia is governed not only by the deck itself but also by the cables and pylons. The concept of “effective mass” can be introduced to give a clearer understanding regarding this point. This is the same concept as the calculation of mass required for sectional model wind tunnel tests.
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Consider the distributed mass of the structure m(x). When the vibration mode is expressed by (x), the generalized mass for this particular mode of vibration for the structure is given by
M " m(x) (x) dx, 1 1 where the integration is to be done not only over the bridge deck but over the whole bridge, including the cables and pylons. Eq. (1) considers a two-dimensional slice of the bridge just like the idea of the conventional sectional model wind tunnel tests, where constant mass per unit length (m) is assumed over the deck length and other bridge components are not considered, with the understanding that the aerodynamic forces on other components are not contributing to the instability. Hence the generalized mass for this idealized system is given by
M " m (x) dx, "Y "Y "Y where m is the required effective mass. The range of integration D refers to the part of the deck represented or the deck length considered. The mode shape is defined by taking the maximum deck displacement as unity. Since two generalized masses, M and M , must be identical, the required magnitude of the effective mass m is 1 "Y decided as follows: m (x) dx# m (x) dx# m (x) dx ! ! ! . . . m"" " " , (2) (x) dx "Y "Y where subscripts D, C and P refer to the deck, cables and pylons. The effective mass m should be the mass parameter m applied in Eq. (1). The effective mass moment of inertia J can be defined in a similar way. The total mass contribution in relation to the amount of energy supplied by the aerodynamic excitation for each mode can be thus expressed by a simple parameter m r /(oB), where r "(J /m ) . The change of this parameter, together with the change of eigen frequencies of the modes which are participating in the flutter instability for the proposed erection procedure of the Ho¨ga Kusten bridge are shown in Fig. 5. The flutter speed is expected to increase in proportion to the square root of this parameter as it appears in Eq. (1). 5.3. Flutter analysis Together with the experimentally obtained data, Fig. 6 shows the calculated flutter wind speeds both from 2D and 3D analyses. The 2D analysis is from Eq. (1). The change of mass, mass moment of inertia and frequency ratio are taken into account in this expression but it is not applicable when the two frequencies are very close to each
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Fig. 5. Development of frequencies and effective mass.
other. The 3D analysis is based on measured aerodynamic derivatives for the Ho¨ga Kusten bridge [4] and the vibration mode shapes given by the dynamic analysis [9]. A flutter condition was not reached for stages 1 and 7 (¸ /¸"0.05 and 0.68) in the 3D analysis.
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Fig. 6. Comparison of critical flutter speeds.
The aerodynamic derivatives were measured using a conventional sectional model for which the aerodynamic interaction is considered to be two-dimensional, or the aspect ratio (span/width) of the bridge is assumed to be infinity. They were also measured in a smooth air stream without any simulation of wind turbulence. However, at very early stages of erection, the assumption of two-dimensionality does not hold, since the aspect ratio is actually very small. For example, the aspect ratio is only 2.75 when ¸ /¸ is 5%. The aerodynamic derivatives for this section were measured separately, using a short span model with open edges, so that the three-dimensional edge effect is involved in the evaluation of the forces. It was found that the characteristics of the aerodynamic derivatives measured in this way were similar to the two-dimensional results, but their magnitude, particularly of the pitching moment components, was significantly reduced due to edge effects. As a result, the flutter speed becomes 5—10% higher. The measurement of aerodynamic derivatives on shorter models and the evaluation of flutter speeds are summarized in Appendix B. The predicted flutter speeds clearly show the tendency of the change in stability limit. The disagreement with the experimental results may be attributed to the effects of turbulence on aerodynamic derivatives which was ignored in their measurement and possible inaccuracy in calculated mode shapes which are different from the original eigen vibration modes once the flutter condition is reached.
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5.4. Effect of non-symmetry in deck configuration The effect of nonsymmetry in deck configuration is more complicated. The comparison of four different deck configurations shown in Fig. 7 and Table 1 is of some help. All four cases, A, B, C and D, have equal deck length, ¸ /¸"0.31, and the cables are rigidly fixed to the deck at the midspan, but the deck eccentricity is D/¸ "0, 0.11, 0.21 and 0.32, respectively. Case A has a symmetric deck and exhibits the lowest frequency ratio and the lowest structural damping. Both bending and torsion are in the first symmetric modes. The eccentricity of deck distribution is increased in the sequence of B, C and D. Note that both frequency ratio and damping are found to increase in this order, contributing to raising the stability limit. For cases C and D, the second torsional mode participated in the flutter whereas the first torsion was coupled in cases A and B. From a review of the mode shapes, it is apparent that the second torsional mode is asymmetric for the cables, but is found to be practically symmetric for the deck, particularly when the deck distribution is eccentric, and the effective mass parameter becomes less for these
Fig. 7. Case of 31% deck with various eccentricity.
Table 1 Dynamic characteristics of 31% deck models Case
D/¸
f (Hz) 4
f (%) 4
f (Hz) 2
f (%) 2
º (m/s) $
f (Hz) $
A B C D
0.00 0.11 0.21 0.32
0.150 0.149 0.146 0.145
0.5 0.9 0.7 0.8
0.204 0.208 0.260 0.237
0.6 0.9 1.5 1.6
57 58 64 65
0.201 0.207 0.221 0.222
Note: f , f — natural frequencies in bending and torsion. 4 2 f , f — structural damping as fraction of critical. 4 2 º , f — flutter speed and flutter frequency. $ $
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mode combinations. When actual instability occurred, the motion was heavily coupled with the lateral sway, also, though the prevailing frequency was close to the torsional frequency. This clearly indicates the fact that the flutter vibration mode cannot be simply described by the eigen vibration modes. The increase of structural damping for complicated mode shapes may be associated with the enhanced distortion of cables and other structural components in real structures. Davenport and Larose [10] have given an interesting explanation of this mechanism for suspension bridge cables. However, how much the structural damping could influence the flutter speed is a different issue. Another interesting case was with 51% of the deck erected. The non-symmetric case with the deck eccentricity of D/¸ "0.13 actually started a fairly violent coupled vibration in bending and torsion at the middle of the span with almost the same onset speed as in the case of the symmetric deck. However, the vibration was observed only locally near the midspan and did not develop to include the entire span. At the further end of the deck, only the lateral sway motion was seen as predominant. It may have been caused by the fact that, as mentioned earlier, the torsional stiffness provided by the cable system is expected to be higher near the pylons compared with the centre of the span. This phenomenon continued at even higher wind speeds until suddenly the divergence-type of destruction occurred. It was an instantaneous catastrophe, and not the gradual increase of vibration amplitude. The deck non-symmetry for this case seems to have increased the post-flutter endurance capacity of the structure.
6. Concluding remarks A series of wind tunnel tests of a suspension bridge with partially erected deck of various lengths was carried out. The flutter stability limit was found to increase when the deck had non-symmetric configurations as opposed to the traditional symmetric distribution. The reason of this phenomenon seems to be very complicated: a combination of vibration modes, frequencies and damping which are all influenced by the deck configurations. Only a three-dimensional modelling of the structure can reveal such behaviour properly. The study also suggested an interesting approach to improve the aerodynamic stability of a bridge during its deck erection. It is possible to choose non-symmetric deck configurations deliberately to maintain a higher stability limit.
Acknowledgements The wind tunnel tests were initially sponsored by the Scandinavian Bridge Joint Venture and its extension was supported by the Danish Maritime Institute (DMI). A. Damsgaard, F.M. Livesey, G.L. Larose and other DMI personnel were involved in the project. The second part of the experiment was largely conducted by P. Reina of
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Politechnico di Milano, whose study at DTU/DMI was made possible through the ERASMUS program, worked out by Dr. C. Dyrbye of the Technical University of Denmark (DTU). Useful discussion was made by Messrs. N. Franck, G. Jensen and B.S. Madsen of Monberg and Thorsen A/S, Copenhagen. The authors extend their appreciation to all of them.
Appendix A. Deflection of a cable under point load Consider a single span cable shown in Fig. 8. It is well known that the deflected shape of the cable is expressed by w¸ m(1!m), y(m)" 2H U
(A.1)
where ¸ is the span, w the weight of the cable per unit length, H the horizontal U component of the cable tension and m"x/¸. In addition to this, if a single concentrated load P is applied at distance a"k¸ from the left support A, the deflection of the cable at the loading point (m"k) is given by w¸ g(m)" bk(1!k), 2H U
b"(1#2j)/c!1,
(A.2)
where j"P/(w¸), c"1#H /H and H the horizontal component of the cable . U . tension due to P. An analytical expression for c is as follows: c(j,k)"[1#12j(1#j) (k!k)] .
Fig. 8. Deflection of a cable under point load.
(A.3)
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When the parameter j is less than 0.5 (say), the following expression can be used with good accuracy: b"0.075(j) /k.
(A.4)
Substituting Eq. (A.3) in Eq. (A.2) and also considering that H "w¸/(8s), s"d/¸ (sag ratio) U
(A.5)
the deflection due to P is given by 0.3Ps (1!k). g(k)" (j) w
(A.6)
Hence the stiffness of the cable in the case of an eccentric point loading is inversely proportional to the length of an increased half of the cable, ¸(1!k). The deck suspended with eccentric distribution would thus have less deflection for the same mass. In the case of torsional oscillations of a partially suspended deck with a certain length, the girder extended closer to one of the pylons is subjected to much higher cable stiffness and, since the deck itself has a large torsional stiffness, the overall stiffness is considered to become quite high.
Appendix B. Aerodynamic derivatives with end effects B.1. Aerodynamic derivatives The aerodynamic derivatives of the Ho¨ga Kusten sectional model were measured with its aspect ratio changed in four steps [11]. The measurement was carried out in the boundary layer wind tunnel of the Danish Maritime Institute by applying the systems identification technique which was developed earlier at DMI [12]. Experimental results are expressed in the dimensionless coefficients defined by Scanlan as follows:
oº hQ BaR h ¸" 2B KH* #KH* #KH*a#KH* , º º B 2
oº hQ BaR h M" 2B KA* #KA* #KA*a#KA* , º º B 2
(B.1)
where ¸ and M are the lift force and pitching moment per unit length of the deck, º the mean wind speed, B and ¼ along-wind and across-wind length of the deck and H*, H*,2, A* are the derivatives given as functions of reduced frequency K"2pfB/º in which f is the frequency (Hz). The aspect ratio of the model (¼/B) was 2, 3, 4 or 5. ¼/B"5 corresponded to the wall to wall or 2D case and hence could be
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Fig. 9. Aerodynamic derivatives, ¼/B"2.
regarded ideally as the same as ¼/BPR. The measurements were carried out in the Reynolds number range of 2;10—4;10. The wind flow was smooth with a longitudinal turbulence intensity of less than 1%. Test results are summarized in Figs. 9—12.
B.2. Stability limits The flutter stability limit for each ¼/B ratio was examined by free vibration tests in turbulent flow. It was also calculated using the measured aerodynamic derivatives above. The results are shown in Table 2. The stability limits in Table 2 are listed in terms of reduced velocity º "º /(Bf ). 0 $ 2 All through the tests, the frequency ratio of torsional to bending motion of the system was kept constant, at f /f "1.55. Also the dimensionless mass and mass moment 2 4
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Fig. 10. Aerodynamic derivatives, ¼/B"3.
parameters, k"m/(oB) and l"J/(oB), were kept constant with values of 17.5 and 2.06, respectively, where m and J are the mass and mass moment of inertia of the model per unit deck length. Hence the results shown in Table 2 can be considered as the effects of having the open ends of the model or the effects of its “threedimensionality”. The calculated flutter stability limits are generally found to be 4—8% lower than the directly measured results. This difference was perhaps caused by two reasons: one is the existence of flow turbulence for the case of direct measurement and another is possible errors in the measurement of the aerodynamic derivatives. The measurement of flutter speed in the dynamic test has less chance of being contaminated by experimental errors. The flow turbulence in the dynamic tests can be approximately characterized by the intensities and scales of I "6.3%, S
I "5.7%, U
V¸ "0.40 m, S
V¸ "0.17 m, U
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Fig. 11. Aerodynamic derivatives, ¼/B"4.
whereas the smooth flow had the turbulence intensity of less than 1%. The flow conditions at the deck level in the Martin Jensen wind tunnel, as a comparison, were I "11.0%, S
I "8.0%, U
V¸ "1.06 m, S
V¸ "0.11 m. U
As it is obvious from these results, the shorter the deck is, the more end effects become predominant. Or it can be considered as a deviation from the conventional concept of strip theory. The aspect ratio ¼/B"2 corresponds to a full-scale deck length of 44 m or ¸ /¸"0.036 for the case of the full bridge model, which is similar to what was tested as the shortest deck. For this case, the expected end effects on the measured flutter speed is of the order of 6% judged from these results.
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Fig. 12. Aerodynamic derivatives, ¼/B"5 (R).
Table 2 Stability limits versus aspect ratio Aspect ratio (¼/B)
2.0
3.0
4.0
5.0 (R)
Measurement (º ) 0 Calculation (º ) 0
7.3 7.0
7.2 6.9
7.1 6.8
6.9 6.3
References [1] F. Brancaleoni, The construction phase and its aerodynamic issues, Proc. Int. Symp. on Aerodynamics of Large Bridges, Copenhagen, February 1992, pp. 147—158. [2] A. Larsen, Prediction of aeroelastic stability of suspension bridges during erection, Proc. 9th ICWE, vol. 2, New Delhi, January 1995, pp. 917—927.
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[3] H.S. Svensson, I. Kovacs, Examples of analytical aerodynamic investigations of long-span bridges, Proc. Int. Symp. on Aerodynamics of Large Bridges, Copenhagen, February 1992, pp. 171—182. [4] F.M. Livesey, Wind tunnel studies of the Ho¨ga Kusten Bridge during construction, Part II: Aeroelastic model tests, DMI Report 94124, Danish Maritime Institute, June 1995. [5] H. Tanaka, A. Damsgaard, P. Reina, N. Franck, B.S. Madsen, Aerodynamic stability of a suspension bridge with partially constructed bridge decks, Presented at 15th IABSE Congress, Copenhagen, June 1996. [6] C. Pedersen, Construction of superstructure of the Ho¨ga Kusten Bridge — suspension bridge in Sweden with a 1210 m main span, Proc. Conf. Bridge into the 21st Century, Hong Kong, October 1995, pp. 243—250. [7] N.J. Gimsing, Cable Supported Bridges: Concept and Design, Wiley, New York, 1983,1996. [8] I.P. Smith, The aeroelastic stability of the Severn suspension bridge, NPL Aero Report 1105, May 1964. [9] A. Damsgaard, Ho¨ga Kusten Bridge dynamic structural analysis, DMI Report 94115, Danish Maritime Institute, April 1995. [10] A.G. Davenport, G.L. Larose, The structural damping of long span bridges: an interpretation of observations, Proc. Canada—Japan Workshop on Bridge Aerodynamics, Ottawa, September 1989, pp. 111—118. [11] P. Reina, The aerodynamic stability of a suspension bridge during erection, M.Sc. thesis, Politecnico di Milano, April 1996. [12] N.K. Poulsen, System identification in determination of the flutter coefficients, Status Report, Skibsteknisk Laboratorium, 1990.