span bridge was simulated by the improved WAWS in this paper while only the deck wind field was usually simulated in the former time-domain analysis.

The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 202 Verification of time-domain buffeting theory and analysis of influence factors for long-span cable-stayed bridges Wanshui HAN a, *, Airong CHEN b, Jiawu LI a,jianxin LIU a a Key Laboratory for Bridge and Tunnel of Shanxi Province, Chang an University, China b State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai, China ABSTRACT: In order to verify the current time-domain buffeting theory, the buffeting-induced structural displacements were measured in a full model test of the Hangzhou Bay Bridge. A time-domain buffeting analysis of aerodynamic forces on bridge deck, towers and main cables was performed to get the buffeting responses, where the aerodynamic admittance function was set as and Sears, respectively, and the equivalent wind spectrum was used to calculate the buffeting loads while considering aerodynamic admittance function. Wind tunnel tests were carried out to study the spatial correlation, and the influences of Sears admittance function and spatial correlation on the buffeting responses were also investigated. The numerical results were compared with those of wind tunnel test. It shows that when the decay factor is 2.9, wind tunnel test results fall between buffeting responses with aerodynamic admittance being set to and Sears, the results calculated with Sears function being on the unsafe side. Spatial correlation greatly influences buffeting responses, the buffeting responses can be reduced by about 3%~22% using the coefficient of spatial correlation obtained in wind tunnel test instead of that proposed in Wind-resistent Design Specification for Highway Bridges. Analysis of the Hangzhou Bay Bridge shows that the current timedomain buffeting theory cannot predict bridge buffeting-induced responses accurately, certain discrepancy exists between the calculation and the tunnel test results. Keywords: cable-stayed bridge; time-domain; buffeting analysis; aerodynamic admittance; Sears function; spatial correlation. Introduction Due to diversity of factors involved in long-span bridges buffeting and comparatively complex theoretical analysis, so far there have been many proposed methods to predict buffeting response of long-span bridges (Aas-Jakobsena and Strommen,200; Scanlan and Gade,977; Scanlan,977; Jain et al,996; Ding,200). Analysis in frequency domain is a conventional approach to predict the buffeting response. However, for a super long-span bridge, where highly nonlinear and coupled responses due to significant wind-structure interaction are encountered, the time-domain approach is more competitive. Long-span bridge nonlinear time-domain buffeting analysis consists of three key components: numerical simulation of wind field, expression of aerodynamic forces and calculation of buffeting responses. The overall stochastic wind field of long 774

span bridge was simulated by the improved WAWS in this paper while only the deck wind field was usually simulated in the former time-domain analysis. In the full model test of the Hangzhou Bay Bridge, the buffeting response of the full model was obtained under the action of the turbulent flow generated in the wind tunnel. Therefore, the analytical results will agree better with the experimental results if the wind-tunnel turbulence is used as the input. For that idea, records of wind tunnel turbulence of the full model test are analyzed to extract its statistical characteristics, which will in turn serve as the target input to the numerical simulation. Spatial correlation is an important factor that influences buffeting responses, however, data from field measurement of spatial correlation of natural wind are generally not available. Decay factor ranges from 7 to 20 and decay factor is set as 7 when there are not data from field measurement of spatial correlation, which is specified in Wind-resistent Design Specification for Highway Bridges. A study on spatial correlation of wind tunnel was carried out in the wind tunnel test. In order to evaluate the effect of the wind spatial correlation on buffeting responses, buffeting response analysis were conducted and decay factor was set as test result, 7 and 20 respectively in the course of wind field simulation. Buffeting analysis has been based on the quasi-steady theory considering Sears admittance function since Sears (Sears,94) put forward the notion of aerodynamic admittance, however, the values calculated based on which are not always agreement with test results. How much the difference will be? Whether the influence of Sears admittance function to each component of buffeting responses (such as RMS of vertical, lateral, torsional displacements) will be similar? All of these will be investigated in this paper.. Turbulence characteristics.. Characteristics of turbulence generated in the wind tunnel Measured horizontal and upward wind spectrum in wind tunnel can be expressed by Eq.(), the measured and fitted auto spectrum of u and w are shown in Fig. and Fig.2. 2 ns cm a / u* af / bf () where, the subscript a of S can be u, v or w, indicating the alongwind, horizontal and upward components of velocity fluctuations of wind turbulence, respectively; c=5/3 for the auto spectra; and a, b, m are the parameters fitted with non-linear least square method based on the measured spectral data. Test data Fitted (a=374.9,b=49.7,m=0.99) Simiu 0. Test data Fitted (a=8.798,b=3.,m=0.9) Panofksy ns u (n)/u 2 * 0. ns u (n)/u 2 * 0.0 0.0 E-3 E-3 E-3 0.0 0. 0 nz/u Fig.. Auto spectra of component u E-3 0.0 0. 0 nz/u Fig.2. Auto spectra of component w 2 775

The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 202 Co-coherence of wind is often expressed by an exponential function: C S uu x, x2, f fd coh exp (2) S x, f S x, f U where, S uu x, f S C uu x x, f uu uu 2 is the power spectrum of along wind-speed fluctuation u at x,, 2 is the real part of cross-spectrum of u between x and x 2, the decay factor, f the frequency, D x x2 the transverse distance and U the mean wind speed. A study on spatial correlation in wind tunnel was carried out in the full model test of the Hangzhou Bay Bridge. Spatial correlation is the function of f and D, In the measurement, D is set as.2 2.24 3.39 and 4.54m respectively, the least square method was used to achieve according Eq.(2), the decay factor ranges from 0.8 to 5.7 and the average of all data is 2.9. 2.3. Results of wind turbulence simulation and checks The improved WAWS was chosen to simulate characteristics of wind field at Hangzhou Bay Bridge (Fig.3). The measured turbulence wind spectra and spatial correlation were used in the simulation. Main parameters in the simulation: K=0.4, z 0 =0.03, U(55.4) = 40m/s; upper cutoff frequency up =4 (rad/s). Figs.4-6 show comparisons of target and simulated auto spectra, coherence function and correlation variance respectively by the improved WAWS. The spectra, coherence function and correlation variance of the simulated turbulence agree well with the input target on the whole. Fig.3. Layout of wind simulation points of Hangzhou Bay Bridge Auto spectrum(m 2 /s) 0 3 0 2 0 0 0 0 - point30 point20 Target Simulated point3 Coherance functions.0 0.8 0.6 0.4 0.2 Target Simulated 0.0 0-2 0.0 0. Frequency(Hz) Fig.4. Auto-spectrum S u 0.0 0. Frequency(Hz) Fig.5. Coherence functions. 3 776

25 0 R 30,30 R 20,30 Correlation variances 20 5 0 5 R 0 30,30 Correlation variances 8 6 4 2 0 R 0 20,30 0-2 -5-400 -200 0 200 400 Time(s) -400-200 0 200 400 Time(s) Fig.6. Comparison of simulated and target correlation variances by the improved method 2. Wind Tunnel Test and Analysis Hangzhou Bay Bridge is a cable-stayed bridge with a span arrangement of 70+60+448+60+70 m, a twin girder steel deck, two inverted Y reinforced concrete pylons. The full bridge aeroelastic model test was carried out in the TJ-3 Boundary Layer Wind Tunnel of the State Key Laboratory for Disaster Reduction in Civil Engineering at Tongji University. The model, shown in Fig.7, was manufactured at a geometric scale of :00. The total length and height of the model are 9.08m and.83m, respectively. Fig.7. Full bridge aeroelastic model in TJ-3 Wind Tunnel 3. Buffeting-induced displacements test and analysis 3.. Interaction between deck, towers and cables In previous time-domain buffeting analyses, only deck wind field was simulated, this primarily owing to buffeting analytical method limitation. In this section, besides the buffeting forces and aeroelastic forces on the bridge deck, the buffeting forces on the towers and the cables are also included in the computation. The results are denoted by the term full bridge. The full-bridge results are then compared with those forces on the bridge deck only, both on the bridge deck and tower, and both on the bridge deck 4 777

The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 202 and cable respectively. In this way, the interaction between the bridge deck, towers, and main cables can be examined. Vertical displacement(m) 0.35 Deck Only Deck and Pylon 0.30 0.25 0.20 0.5 0.0 0.05 lateral displacement(m) 0.050 0.045 0.040 0.035 0.030 0.025 0.020 0.05 5 Deck Only Deck and Pylon angular displacement( o ) 0.8 Deck Only 0.6 Deck and Pylon 0.4 0.2 0.0 0.08 0.06 0.04 0.02 0 Fig.8. Comparisons of displacement responses RMS values at mid-span Fig.8 shows the variable curves of mid-span s vertical, lateral and angular displacement RMS values for the full bridge, for the deck only, for both of the deck and tower, for both of the deck and cable and the test values. Table presents comparisons of midspan s lateral displacement RMS values for the full bridge, for the deck only the test values. As shown in the Fig.8 and Table, conclusions can be drawn as follows: ), The buffeting forces on the cables and towers have a mild effect on the vertical and torsional displacement responses of the bridge deck at mid-span. 2), The buffeting forces on the towers contribute little to the lateral buffeting-induced responses of bridge deck. 3), Due to the consideration of buffeting forces on the main cables, the lateral displacement response of the bridge deck increases greatly. Take designed wind speed of 50m/s as an example, the full bridge lateral displacement response RMS value is 0.034m, 3.8% larger than the value of 0.08m if only taking deck wind field into consideration. Moreover, the differences become larger when wind speed increases. Therefore, the cable wind field must be taken into consideration, especially in the buffeting analysis of super long-span cable-stayed bridges whose cables are very long. 4) The displacement deck at midspan are a little larger than wind tunnel test results and agree well with each other on the whole, especially for lateral and torsional displacement RMS values. For vertical displacement mid-span, calculated values and test values have a good accordance at low wind speeds. The calculated vertical displacement RMS value at mid-span is.7 times larger than test values at design wind speed of 50m/s, whereas the difference become larger when wind speed increases. longitudinal displacement(m) 0.2 0.0 0.08 0.06 0.04 0.02 Deck Only Deck and Pylon lateral displacement(m) 0.035 0.030 0.025 0.020 0.05 5 Deck Only Deck and Pylon Fig.9. Comparison of displacement response RMS values at the top of pylon 0 5 778

Some conclusions can be found from Fig.9: ), The buffeting forces on the cables have a little influence on lateral displacement at the top of pylon. At designed wind speed of 50m/s, the lateral displacement response RMS value with buffeting forces on the deck and cables is 45m, 9.7% larger than the value of 4m if only taking deck wind field into consideration. 2), The buffeting forces on the towers have a considerable influence on the lateral displacement RMS values. At designed wind speed of 50m/s, the full bridge lateral displacement response RMS value is 84m, 04.8% larger than the value of 4m if only taking deck wind field into consideration. 3) The full-bridge lateral displacement response RMS values at the top of pylon agree with the test values well while those forces on the bridge deck only are much smaller than the tested values. Therefore, ignorance of buffeting forces on pylon and cable will achieve insecure results. Spectral amplitude(m 2 /sec) 0. 0.0 E-3 E-4 E-5 E-6 Deck only Spectral amplitude(m 2 /sec) 0.0 Deck only E-3 E-4 E-5 E-6 E-7 E-7 0. Frequency(Hz) E-8 0. Frequency(Hz) Fig.0. Comparison of power spectrum density of displacement response at the top of pylon Table Main modes of Hangzhou Bay Bridge Mode Frequency Frequency Mode type Mode (Hz) (Hz) Mode type 0.86 LF 8 0.87337 A-L- 2 0.3685 S-V- 9 0.876 A-V-2 3 0.47598 S-L- 5 0.9533 S-T- 4 0.5054 A-V- 7.02839 S-V-3 5 0.53307 A-L(Tower) 23.6229 A-V-3 6 0.56004 S-L-2 24.4749 A-T- 7 0.72903 S-V-2 25.45489 S-V-4 Note:S=Symmetrical; A=Antisymmetrical; LF=Longitudinal Floating; V=Vertical; T=Torsion; L=Lateral. The spectra of the lateral and longitudinal displacement responses at the top of pylon are shown in Fig.0. The longitudinal displacement response spectra for the full bridge are almost agreement with the deck only. Five peak values occur in each power spectrum density curve VS the frequency of mode 2, 4, 7, 9, and 7. Moreover, the mode 6 779

The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7) Shanghai, China; September 2-6, 202 types of these five peak values are all vertical symmetric bending or vertical dissymmetric bending of deck (Table 3). Therefore, can conclude that the longitudinal vibration of pylon mainly depend on the vertical vibration of deck. The lateral displacement response spectra for the deck only are obviously smaller than those for the full bridge. The former three peak values occur in each power spectrum density curve VS the frequency of mode 3, 5 and 6. The type of mode 5 is a dissymmetric lateral bending of pylon, but the type of mode 3 and 6 are both symmetric lateral bending of deck. Compared with the case only considering deck wind field, the contribution of mode 5 becomes more significant, and the contributions of mode 3 and 6 become weaker when taking full bridge into account. 3.2. Influence of aerodynamic admittance Since Sears established the conception of aerodynamic admittance in 940s, buffeting analysis is always carried out based on quasi-steady theory considering Sears admittance function. The difference in the actual value and the predicted value adopting Sears admittance function can t be ignored in practical projects. But how much the difference will be and how Sears function influence the displacement response (such as vertical, lateral and torsional displacement response RMS values) are still unknown to us. Therefore some researches are needed to resolve the above problems and are presented in this paper also. Vertical displacement(m) 0.35 0.30 0.25 0.20 0.5 0.0 0.05 Admittances: Admittances:Sears lateral displacement(m) 0.050 0.045 0.040 0.035 0.030 0.025 0.020 0.05 5 Admittances: Admittances:Sears angular displacement( o ) 0.8 0.6 0.4 0.2 0.0 0.08 0.06 0.04 0.02 Admittances: Admittances:Sears 0 Fig.. Comparison of displacement response RMS values at mid-span Fig. show the buffeting responses at mid-span with and without consideration of Sears aerodynamic admittance. Some conclusions can be obtained thereinafter: ), The RMS values considering Sears function are smaller than those without considering admittance function and the ones in wind tunnel. 2), The influence of Sears admittance function to each component of buffeting-induced displacement responses is different. Compared with vertical and lateral displacement responses, the ratios of torsional RMS values considering Sears admittance function to those without considering admittance function and the ones in wind tunnel are much smaller. 5. Conclusions ) The buffeting forces on the cables and towers have a little effect on the vertical and torsional displacement responses of the bridge deck and the longitudinal responses of tower. The buffeting forces on the cables contribute mainly to the lateral responses of bridge deck while the buffeting forces on the towers can increase the lateral responses 7 780

of pylon significantly. Therefore, ignorance of buffeting forces on pylon and cable will achieve insecure results. 2) The calculated buffeting RMS values are slightly larger than wind tunnel test results and agree with the wind tunnel test results on the whole when admittance function is and is 2.9. It is suggested that buffeting RMS values when admittance function is and the value of is from field measurement can be served as controlling values in wind-resistance design when appropriate admittance function is not available. Acknowledgements The writers are grateful for the financial support from the National Natural Science Foundation of China under Grant NNSF-5047809. References [] Aas-Jakobsena, K., Strommen, E.,200. Time domain buffeting response calculations of slender structures. Journal of Wind Engineering & Industrial Aerodynamics 89, 34-364. [2] Bucher, C. G., Lin, Y. K., 988. Stochastic stability of bridges considering coupled modes.journal of Engineering Mechanics 4(2), 2055-207. [3] Chen, X., Kareem, A., 200. Equivalent static wind loads for buffeting response of bridges. J. struct. Engrg. 27(2), 467-475. [4] Chen, A. R., Han, W. S., 2004. Technical report of wind tunnel study on wind-resistant performance of Hangzhou Bay Bridge. State Key Laboratory for Disaster Reduction in Civil Engineering (In Chinese). [5] Communication Department of P. R. of China, 2004. (JTG/T D60-0-2004) Wind-resistant Design Specification for Highway Bridges, Beijing, China Communications Press (In Chinese). [6] Deodatis, G., 996. Simulation of ergodic multivariate stochastic processes. J. Engrg. Mech. 22 (8):778-787. [7] Ding, Q. S., 200. Refinement of coupled flutter and buffeting analysis for long-span bridges, Ph. D. Dissertation, Tongji University, Shanghai, China (In Chinese). [8] Ding, Q. S., Chen, A.R., Xiang, H.F., 2006. Simulation of spatial fluctuating wind field on long span bridges. Chinese quarterly of Mechanics 27 (2):84-89 (In Chinese). [9] Homles, J. D., 2002. Effective static oad distributions in wind engineering. J. Wind Engrg. Indust. Aerodyn. 90, 90-09. [0] Jain, A., Jones, N. P., Scanlan R. H., 996. Coupled aeroelastic and aerodynamic response analysis of long-span bridges. Journal of Wind Engineering & Industrial Aerodynamics 60, 69-80. [] Sears, W. R., 94. Some aspects of non-stationary airfoil theory and its practical application.journal of Aeronautical Sciences 8, 04-08. [2] Scanlan R. H., Gade R. H.,977. Motion of suspended bridge spans under gusty wind. Journal of Structure Engineering 03 (9), 867-883. [3] Scanlan R. H., 978. The action of flexible bridges under wind, II: buffeting theory. Journal of Sound and Vibration 60(2), 20-2. 8 78