of a thin airfoil in fully correlated gusts with sinusoidal fluctuations(liepmanm 195): χ L * ( f ) 1 1 π f = + (1) where is the square of modules of

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1 Shanghai, China; September -6, 01 Measurements of equivalent aerodynamic admittances of two separate and parallel bridge decks Qi Zhou a,b, Le-Dong Zhu a,c, Peng-Jie Ren a,c a State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China b Department of Civil Engineering, Shantou University, Guangdong, China, c Key Laboratory for Wind Resistance Technology of Bridges of Ministry of Transport, Tongji University, Shanghai, China ABSTRACT: The concept of aerodynamic admittance functions was introduced into buffeting analysis of long-span bridges to consider the unsteadiness and spatial variation of wind turbulence and plays an important role in improving the accuracy of buffeting response. Because of the signature turbulence effect and aerodynamic interference effect, the aerodynamic admittance of bridge with two separated paralleled decks may no longer behave as Sears function. Taking Tanggu Haihe Bridge in Tianjin as an engineering back ground, the aerodynamic admittances of a semi-closed box deck a full-closed box deck were measured and analyzed in this article. Moreover, the aerodynamic interference effect on the equivalent aerodynamic admittance with different types of bridge decks at the windward or leeward position and with different distances between two decks were also investigated via wind tunnel test by the method of force measurement. The results show that the equivalent aerodynamic admittances of windward and leeward deck are quite different from that of single deck in their characteristics due to the aerodynamic interference effect. The aerodynamic interference effect on the curve is more significant than those on the and curves, and that on the leeward deck is much stronger than that on the windward deck. Furthermore, with variation of the ratios of D/B (the ratios of distance between two decks over the deck width), the aerodynamic interference may exert different effects on the equivalent aerodynamic admittances. For both of the of the windward and leeward decks, the admittance curves approach to that of the single deck gradually with the increasing reduced frequency. And for the and, there is no significant difference between the admittance curves of the windward deck and the single deck, whilst that of leeward deck are remarkable different from that of single deck, especially for the curves. KEYWORDS: long-span bridge; wind tunnel test; aerodynamic admittance; signature turbulence effect; aerodynamic interference effect 1 INTRODUCTION Buffeting is one of important types of wind-induced vibration of long-span bridges due to their flexibility and lower fundamental natural frequencies, and has been paid great attentions to in both wind engineering and bridge engineering fields. Up to now, apart from wind tunnel test means, various analysis methods have been developed for predicting buffeting responses of long-span bridges. The Davenport buffeting theory and linear self-excited aeroelastic force model introduced by Scanlan and his co-workers were generally accepted in the above analysis methods. In Davenport buffeting response prediction, Quasi-steady theory was employed to establish the buffeting forces on the bridge and aerodynamic admittance functions were introduced to consider the unsteadiness and spatial variation of wind turbulence surrounding bridge deck cross sections. The aerodynamic admittance can either be approximated as in Liepmanm(195),Davenport(196), Irwin(1977), or measured as in Holmes(1975), Kawatani & Kim (199), Sankaran & Jancauskas(199), Larose(199) and Bogunovic Sears function is most commonly used form of the lift aerodynamic admittance 78

2 of a thin airfoil in fully correlated gusts with sinusoidal fluctuations(liepmanm 195): χ L * ( f ) 1 1 π f = + (1) where is the square of modules of equivalent aerodynamic admittances of lift; f * is a reduced frequency = fb U, U is the mean wind speed, and B is the whole width of the deck in this paper. However, the buffeting forces in most of the current buffeting analysis are determined based on the quasi-steady theory, where, only the turbulence of incident wind is included. The signature turbulence caused by the interaction between structure and flow around it is normally ignored. Up to the present, the extent of signature turbulence effect on the bridge buffeting responses has not been understood yet and has rarely been investigated, although this issue was already regarded as one of major challenges in the prediction of long-span bridge response to wind (Jones 1999). With the development of traffic flow, bridge with two separated paralleled decks has been built to accommodate the traffic requirement in long-span bridge engineering field, and it has been observed that aerodynamic interference phenomena exist between the two neighboring decks. And the interference behavior largely depends on the distance between the two decks (Honda et al., 1993; Larsen et al., 000; Stoyanoff et al., 003; Kimura et al., 008). If the two bridge decks are close enough to each other, the surrounding flows around one bridge deck will be influenced by another, and vice versa. Consequently, the performance of cross-sectional aerodynamic admittance of each deck will be inevitably and significantly affected by another deck. However, Most of the conducted researches referred to above focused extensively on revealing the complicated behavior of the aerodynamic interference phenomenon, little research has been conducted to aerodynamic interference effect on the aerodynamic admittances, which may make significant effect on the buffeting response. In this connection, the equivalent aerodynamic admittances are investigated in this study via force measurement wind tunnel tests of sectional model in turbulence flow. Because the signature turbulence is regarded to be much more significant for bridge decks with separate boxes than other types of decks, the existing and new Tanggu Haihe Bridge in Tianjin(see Figure 1), which are both cable-stayed single tower bridge with a main span of 310, are thus taken as a engineering back ground in this study. Furthermore, the aerodynamic interference effect on the equivalent aerodynamic admittances of bridge with two separated paralleled decks are also surveyed in these wind tunnel tests, which adopted seven D/Bs(the ratio of distance between two decks and deck width) conditions and six cases with the combinations of different type decks. The details of the research are to be introduced in the following sections. * 35 Elevation (m) Existing Br. New Br. Haihe River 35 Figure 1 Layout of the existing and new Tanggu Haihe Bridge in Tianjin (Unit: m) 783

3 Shanghai, China; September -6, 01 DESCRIPTION OF SECTIONAL MODEL TEST The motionless sectional model tests for the equivalent aerodynamic admittances of the bridge deck were carried out in the TJ- Boundary Layer Wind Tunnel at a length scale of 1/60 and with turbulent intensities of 15%. In these wind tunnel tests, two types of sectional models, designed based on the existing and new Tanggu Haihe Bridge decks, were with the shape of semi- and full-closed box deck respectively(see Figure ). The case simulating real Tanggu Haihe Bridge, full-closed box deck at windward and semi-closed box deck at leeward, is called as case FSD for short in this paper, and called as case SFD vice versa. Another two cases, two semi-closed box deck(case SSD) and two full-closed box deck(case FFD), are also investigated in order to reveal the relationship between aerodynamic interference effect and distance between two decks. And D/Bs of 0, 0.5,0.5,0.75,1,1.5, were adopted in these tests. Moreover, in order to compared with the cases mentioned above, two cases were also measured in additionally, and the case with single full-closed box deck is named as case FD, and likewise the case with single semi-closed box deck is called case SD in this paper D Aluminum Rectangle Bar Full-closed Sectional Model Semi-closed Sectional Model Figure Measured segment of sectional model with semi- and full-closed box deck (Unit: mm) Figure 3 Sectional model in TJ- Boundary Layer Wind Tunnel Figure 3 shows the sectional model mounted in the TJ- wind tunnel. The sectional model was designed in such a way that the fluctuating forces could be synchronously measured in the test for the windward and leeward deck. Therefore, the sectional model was comprised of two measured segments and two upper compensatory segments. The windward segment and its upper compensatory segment were simulated the aerodynamic shape of windward bridge deck, and so does the leeward segment. The measured segment was made of wood and an aluminum link endplate, which was used to connect the measured segment to the balance. The measured segment of semi-closed box deck was 0.4m long (excluding the end plate), 0.05m high and 0.44m wide, and that of full-closed box deck also 0.4m long (excluding the end plate) and 0.05m high, but 0.41m wide. The total mass of the measured segments with 784

4 semi- and full-closed box are 1.01kg and 1.4kg, respectively. The aluminum end plate had a basic shape as same as the cross section shape of the measure segment, but its middle part was circular with a diameter as same as the force balance. Two upper compensatory segments were also 0.4m long and were made of same material, but empty inside for reducing weights. Before the test, a steel frame for the installation of compensatory segments was mounted on the turntable at first. Two five-component force balances were then fixed separately on sectional steel bars with sliding grooves, and their surfaces was 0.3m upon the tunnel floor. Afterwards, to depress the disadvantageous effects of boundary layer above the tunnel floor and 3D flow around the lower end of the sectional model, a rectangular separating plate made of Perspex was then mounted upon the turntable with its bottom surface level a little bit lower than the top surface level of the force balance. In the middle region of the separating plate, there was an opening with a shape similar to the aluminum end plate of the measured segment, but a little bit larger than the latter. The separating plate must keep no touch to the top perceptional surface of the force balance. In the next step, two measured segment were vertically mounted on the force balances through the preset opening of the separating plate without any contact, and two upper compensatory segments were installed on the steel frame at their top ends and were kept a narrow gap of 1-mm from the measured segment at their bottom ends. Finally two upper compensatory segments were connected to each other through the transverse link beam model to increase their stiffness. As a result, the fundamental natural frequencies of the measured segment with semi-closed box were 33Hz, 58Hz in the directions in and out of the deck plane(fx and Fy), respectively, and 8Hz in torsional direction(mz). Consequently, the fundamental natural frequencies of the measured segment with full-closed box were 33Hz of Fx, 50Hz of Fy and 71Hz of Mz. Both of the frequencies mentioned above were much higher than 15Hz calculated by the third vertical bending frequency of Tanggu Haihe Bridge and the frequency ratio of wind tunnel test. 3 RESULTS OF EQUIVALENT AERODYNAMIC ADMITTANCES The spectra of buffeting drag (S DD ), lift (S LL ) and torsion moment (S MM ) on the bridge deck can be expressed with the following equations according to the quasi-steady theory (Chen et al. 001): ' ( ρ ) 4 χ ( ) χ ' * * * + CD( CL + CD) ( χduχdwsuw + χduχdwsuw) ' ' ( ρ ) 4 4 ( ) ( ) S = UB C S + C + C S DD D Du uu L D Dw ww D uu D L D uw L D ww χd = UB C S + C C + C C + C + C S () ρub ' LL = 4 L χlu uu + D + L χlw ww ( ) S C S C C S ' * * * + CL( CD + CL) ( χluχlwsuw + χluχlwsuw) L uu L D L uw D L ww χl ' ' ( ρ ) 4 4 ( ) ( ) = UB C S + C C + C C + C + C S ρub ' MM = 4 M χmu uu + M χmw ww S C S C S ( ρ ) ' * * * + CM CM ( χmu χmwsuw + χmu χmwsuw) ' ' M uu M M uw M ww χm = UB 4C S + 4C C C + C S (3) (4) where the superscript * indicates the conjugate operation; ρ is the density of air; C D, C L and 785

5 Shanghai, China; September -6, 01 C M are the aerodynamic coefficients of mean drag, lift and torsion moment, respectively; C D = dcd / dα, C = dc / d L L α ; χ Du, χ Dw, χ Lu, χ Lw, χ Mu and χ Mw are the equivalent aerodynamic admittances, which are the function of, respectively; ω is circular frequency; S uu and S ww are auto spectra of fluctuating wind components u and w, respectively; S uw and C uu, are the cross spectrum and co-spectrum between u and w, respectively;, and are the square of modules of equivalent aerodynamic admittances of drag, lift and torsion moment, respectively, considering the joint contribution of u and w. Figure 4 shows the results of, and of case FSD measured in the turbulence flow field with the mean attack angle of zero. The measured data are then fitted using the following target functions of fraction series of Eq.(5) for the incident turbulence contribution and of Eq.(6) for the signature turbulence contribution, respectively, and the corresponding fitted curves are also plotted in Figure 4. α χl = (5) r 1 + β K χ c + c K n 1i i s = i= 1 ( K c3i) + c4i (6) Measured(windward deck) Measured(leeward deck) Fitted for the incident turbulence(windward deck) Fitted for the incident turbulence(leeward deck) Fitted for the signature turbulence(windward deck) Fitted for the signature turbulence(leeward deck) Fitted for the total turbulence(windward deck) Fitted for the total turbulence(leeward deck) Sears function Figure 4 Measured and fitted equivalent aerodynamic admittances(eaa) of case FSD From Figure 4, it can be seen that whatever windward deck or leeward deck, the tendencies of the measured equivalent aerodynamic admittances are not similar to that of Sears function. Due to the effect of signature turbulence, the and curves don t decrease with the increase of the reduced frequency, but generally, the curve still follows the reducing tendency as Sears function. In additional, the measured equivalent aerodynamic admittances are generally smaller than Sears function for K<1, and for K>1, the measured equivalent aerodynamic admittances of and are mostly somewhat much lager than Sears function whilst the of windward deck is close to Sears function and the of leeward deck is almost a bit little than Sears function. Furthermore, compared measured equivalent aerodynamic admittances of windward and leeward deck, the of windward and leeward deck have a similar characteristic that there are more than three significant peaks

6 at the range of K>3. Although the and of leeward deck are obviously influenced by the signature turbulence effect, that of windward almost does not affected by the signature turbulence. Besides, it is found that for windward deck, the effect of signature turbulence makes the curve of having three significant peaks at about K=.9, 3.5&3.8, and the curve of having two somewhat peaks appreciable peaks at 4&5, but doesn t influence the curve. For the leeward box, the effect of signature turbulence exerts remarkable influence on all the three equivalent aerodynamic admittances within the reduced frequency zone between.5 and 6, and makes the curve of having four peaks with a value much higher than 1.0 at about K=.9, 3.5,3.8 and 4.8, and influences the and curves of having a peak at about K=4.5 and 5, respectively. According to predecessors research, the equivalent aerodynamic admittance of the signature turbulence is regarded to the aerodynamic shape of bridge deck and the inflow wind spectrum, Therefore, the equivalent aerodynamic admittances of incident and signature turbulence are fitted separated by different formula(eq.(5) and Eq.(6), respectively) in Figure 4, and the result fitted for total turbulence is the sum of that of incident and signature turbulence. Layer1 Layer4 Layer Layer5 Layer3 Layer6 EAA of case FD EAA of case SD windward deck EAA of case FSD windward deck EAA of case SFD windward deck EAA of case SSD Figure 5 Comparison of equivalent aerodynamic admittances(eaa) of windward deck between different cases 4 EFFECT OF AERODYNAMIC INTERFRENCE 4.1 Discussion of aerodynamic interference effect between different cases Figure 5 shows the comparison of the measured equivalent aerodynamic admittances of 787

7 Shanghai, China; September -6, 01 windward deck between different cases. There are six layers of Figure 5 in which layer 1~3 present the measured, and of windward deck in case FD, SD, FSD and SSD and layer 4~6 show that in case SD, SFD and SSD, respectively. From layer 1~3, compared between case FD and FSD or SD and SSD, it is found that the value of windward deck(case FSD or SSD) is remarkable smaller than that of single deck(case FD or SD), but and of windward deck are close to that of single deck. And there are three peaks of curve within case FSD whilst only two remarkable peaks within case FD, which can be explained that the signature turbulence contains more eminent reduced frequencies and wide range of affected frequency as a result of the aerodynamic interference effect. Furthermore, the curves of case FSD and SSD present similar peaks, thus it can be concluded that the equivalent aerodynamic admittance of different windward decks setting a same semi-closed box deck at leeward possess the alike signature turbulence effect. From layer 4~6, it is noticed that with different decks at leeward, the and curves of windward deck are approximate to each other, but for the curve, the semi-closed box deck at leeward exerts much more complicated signature turbulence effect than the full-closed box deck does. Therefore, we can concluded that leeward deck possessed significant signature turbulence effect will make more obvious aerodynamic interference effect on the windward deck. 10 Layer7 Layer10 10 Layer8 Layer Layer9 Layer1 EAA OF case FD EAA OF case SD leeward deck EAA of case FSD leeward deck EAA of case SFD leeward deck EAA of case SSD Figure 6 Comparison of equivalent aerodynamic admittances(eaa) of leeward deck between different cases Figure 6 shows the comparison of the measured equivalent aerodynamic admittances of leeward deck with different cases. There are six layers of Figure 6 in which layer 7~9 present the measured, and of leeward deck in case FD, SD, SFD and SSD and layer 10~1 show that in case SD, FSD and SSD, respectively. From layer 7~9, the equivalent aerodynamic admittances of leeward deck are remarkable different from that of single bridge 788

8 deck. For the and curves, there is no obvious peak in case FD/SD but a distinct peak at the approximation of K=5 in case SFD/SSD. For the curve, there is only one peak in case FD/SD but more than three peaks in case SFD/SSD. Consequently, the signature turbulence effect exerts a wider range of effect reduced frequency in case SFD/SSD than that in case FD/SD. Therefore, it can be further concluded that the equivalent aerodynamic admittance of bridge deck will greatly affected by the aerodynamic interference phenomenon caused by the windward deck. Moreover, compared with case SFD and SSD, because of a same bridge deck at windward, the aerodynamic interference effect on different leeward decks possess a similar behavior. From layer 10~1, it is found that there is no obvious difference of the and curves of leeward deck between case FSD and SSD, but remarkable difference of the curve between case SD and FSD, SSD in high reduced frequency range. Because the semiand full-closed box deck has a same width and height, the aerodynamic interference effects on the drag and torsional moment equivalent aerodynamic admittances are similar to each other. But for different bottom plates of decks, there is a remarkable difference of the lift equivalent aerodynamic admittance. The results above indicate that the same aerodynamic shape of bridge deck at windward will makes a similar aerodynamic interference effect on bridge deck at leeward, and vice versa. Layer1 Layer4 Layer Layer5 Layer3 Layer6 EAA of case SD EAA of case SSD with D/B of 0 EAA of case SSD with D/B of 0.5 EAA of case SSD with D/B of 0.5 EAA of case SSD with D/B of 0.75 EAA of case SSD with D/B of 1 EAA of case SSD with D/B of 1.5 EAA of case SSD with D/B of Figure 7 Comparison of equivalent aerodynamic admittances(eaa) of windward deck with different D/Bs 4. Discussion of aerodynamic interference effect with different D/Bs Figure 7 shows the comparison of equivalent aerodynamic admittances of windward deck with different D/Bs of case SSD and that of case SD. In order to picture clearly, Figure 7 is 789

9 Shanghai, China; September -6, 01 plotted out to six layers. Layer 1~3 present the measured results of windward deck in case SSD with D/B from 0 to 0.75 and the result of case SD, and layer 4~6 list that of leeward in case SSD with D/B from 1 to and the result of case SD. From the curves in layer 1 and 4, it is interesting noticed that the drag equivalent aerodynamic admittance of windward deck with D/B of 0 is a little larger than that of other conditions. Compared with the results of case SD, the curves of case SSD are gradually approaching to that of case SD with the D/B increasing from 0 to. This variation indicates that the aerodynamic interference effect on the of the windward deck will become inconspicuous with the decreasing of the distance between windward and leeward deck. But in layer and 5, we can found that the curves don t follow the change rule mentioned above. Meanwhile, compared with the of case SD affected by the signature turbulence inconspicuously, that of case SSD present somewhat signature turbulence effect with D/B>1. For the of case SSD in layer 3 and 6, there is no obvious difference in the comparison with case SD. Therefore, for windward deck, the drag equivalent aerodynamic admittance will get close to that of single deck gradually with the distance increasing, although it is a nonlinear change, but the lift and torsional moment equivalent aerodynamic admittances present no visible variety with the distance changing. Layer7 Layer10 Layer8 Layer Layer9 Layer1 EAA of case SD EAA of case SSD with D/B of 0 EAA of case SSD with D/B of 0.5 EAA of case SSD with D/B of 0.5 EAA of case SSD with D/B of 0.75 EAA of case SSD with D/B of 1 EAA of case SSD with D/B of 1.5 EAA of case SSD with D/B of Figure 8 Comparison of equivalent aerodynamic admittances(eaa) of leeward deck with different D/Bs Figure 8 shows the comparison of equivalent aerodynamic admittance of leeward deck with different D/Bs of case SSD and that of case SD, and is also plotted out to six layers. Layer 7~9 present the measured results of leeward deck in case SSD with D/B from 0 to 0.75 and the result of case SD, and layer 4~6 list that of leeward in case SSD with D/B from 10 to 1 and the result of case SD. From layer 7 and 10, the curves of case SSD also keep the change serially like that of windward deck with D/B increasing, but the signature turbulence 790

10 effect makes it much more nonlinear and complicated. In layer and 5, due to intensely signature turbulence effect, the aerodynamic interference effect becomes disorderly and makes the curves greatly different from that of case SD. Consequently, although the distance equals zero in case SSD with D/B of 0, the signature turbulence still exerts the curves having more than two peaks at the range of K>3. In layer 9, the aerodynamic interference makes the curves of case SSD with D/B from 0 to 0.75 significant smaller than that of case SD. However in layer 1, the curves of case SSD with D/B>1 are almost equal to that of case SD which means that the aerodynamic interference effect can be ignored with the distance larger than a certain value. In general for leeward deck, with the change of distance between windward and leeward deck, the aerodynamic interference on the follows a certain variation like that of windward. presents disorderly and unsystematically on the. Otherwise, the aerodynamic interference effect exerts the smaller than that of case SD with D/B<1, but can be ignored while D/B>1. 5 CONCLUDING REMARKS The equivalent aerodynamics admittance of bridge with two separated paralleled decks are discussed in this paper. According to the discussion above, some conclusions can be drawn as follows: The aerodynamic admittance curves don t always decrease with the reduced frequency increasing, which is different from the Sears function. Especially in high reduced frequency area, the signature turbulence effect exerts the aerodynamic admittance curves of windward deck of having some significant peaks, and makes that of leeward deck of having much more remarkable peaks. Otherwise, the measured equivalent aerodynamic admittances are generally smaller than Sears function for K<1, but much larger than Sears function at high reduced frequency due to the signature turbulence effect. The aerodynamic interference phenomenon takes a great effect on windward and leeward deck and make their equivalent aerodynamic admittances tremendous different with that of single deck. The aerodynamic interference effect on the curve is serious than that on the and curves, and the aerodynamic interference effects on leeward deck are much more stronger than that on windward deck. Moreover, while the distance between windward and leeward deck changes, the aerodynamic interference may generate different effects on the equivalent aerodynamic admittances. For both the drag aerodynamic admittances of windward and leeward deck, the curves may approach to that of single deck gradually with the D/B increasing. For the lift and torsional moment aerodynamic admittances of windward and leeward deck, the and curves of windward deck don t behave any extraordinary difference from that of single deck, but that of leeward deck are quite different from that of single bridge especially the curves. 6 ACKNOWLEDGEMENTS The work described in this paper was jointly supported by the Fundamental Research Fund for State Key Laboratories from the Ministry of Science and Technology of China (Grant No. SLDRCE08-A-0), and the National Nature Science Foundation of China (Grant ). Any opinions and concluding remarks presented here are entirely those of the writers. REFERENCES A.G. Davenport, Buffeting of a suspension bridge by storm winds. Journal of the Structural Division, 88(196,ST3), A. Honda, N. Shiraishi, M. Matsumoto, et al, Aerodynamic stability of Kansai International Airport access bridge. J. Wind Eng. Ind. Aerodyn. 49(1993), X. Chen and M. Matsumoto, Multimode coupled flutter and buffeting analysis of long span bridges, J. Ind. Aerodyn., 89(001),

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