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Wind-induced response of a cable-stayed bridge under partially and fully correlated buffeting forces
Adrián Pozos-Estrada1, Roberto Gómez-Martínez1, H.P. Hong2, Héctor Hernández-Landero1 1Coordinación de Mecánica Aplicada, Instituto de Ingeniería, Universidad Nacional Autónoma de México, México 2Department of Civil and Environmental Eng., The University of Western Ontario, London, Ontario, Canada email: [email protected], [email protected], [email protected], [email protected]
ABSTRACT: Cable-stayed bridges are sensitive to buffeting forces, and their study is usually carry out analytically and/or experimentally. In the first approach mathematical models and frequency or time domain analysis are carry out to estimate the mean and peak response of interest. In the second approach, scaled section or aerosleatsic models are tested in boundary layer wind tunnels under different types of terrain to calculate the response. If the first approach with frequency domain analysis is adopted, buffeting forces should be characterized. To account for spanwise correlation, a decay function is usually adopted. The main objective of this work is to study the wind-induced response of the tallest cable-stayed bridge in Mexico under partially and fully correlated buffeting forces. For the parametric analyses, a mathematical model of the bridge is developed in Ansys Parametric Design Language (APDL). A harmonic analysis is employed to carry out the buffeting analyses with partially and fully correlated wind forces. The results of the analyses indicate that spanwise correlation affects the peak response of the structure and an evaluation of the amount of correlation should be further study to take it into account in design.
KEY WORDS: Buffeting forces; Wind-induced response; Partial and full correlation; Cable-stayed bridges.
1 INTRODUCTION Wind effects on long span bridges are usually studied analytically and/or experimentally. In the first approach, wind forces are separated into static, buffeting and aeroelastic, and applied to a mathematical model of the bridge that includes all the relevant information to carry out the analyses. For the mean response of the bridge, the mathematical model of the bridge is subjected to static forces along all the elements that contribute to the static response, including the cables in some cases [1]. The dynamic response of the bridge under buffeting forces is commonly separated into a background and resonant response [2], these dynamic responses can be calculated separately by methods based on random vibration, as those proposed in [3, 4]. In other cases, the dynamic response of the bridge can be calculated in the time domain with a mathematical model, if the time histories of the buffeting forces are known, or in the frequency domain, if the buffeting forces are characterized by the corresponding power spectral density functions (PSDF). The experimental approach considers a scaled model of the bridge (section model or aeroelastic model) to study the response of the bridge under wind and to identify possible instabilities. According to [5], the benefit of testing an aeroelastic model is greater than that of a section model; however, the cost of the former can be several times that of the later. For milestone projects of bridges, it is common to design a section and an aeroelastic model. The main objective of this work is to study the wind-induced response of the tallest cable-stayed bridge in Mexico under partially and fully correlated buffeting forces. For the parametric analyses, a mathematical model of the bridge is developed in Ansys Parametric Design Language (APDL), and a harmonic analysis is employed to carry out the buffeting analyses with partially and fully correlated wind forces. The correlation of the buffeting forces is taken into account through the coherence function given by Davenport [6].
2 BUFFETING FORCES ON A BRIDGE Buffeting forces vary in time and space along the length of the deck of the bridge. If the quasi-steady assumption is adopted, the buffeting forces per unit length can be expressed as,
(1a) 1 2 2u(t) ' w(t) Db (t) U BCD CD 2 U U
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(1b) 1 2 2u(t) ' w(t) Lb (t) U BCL (CL CD ) 2 U U
(1c) 1 2 2 2u(t) ' w(t) M b (t) U B CM CM 2 U U where is the air density; U is the mean wind velocity at deck height, CD, CL and CM are the drag, lift and moment coefficients that depend on the angle of attack , C’D, C’L and C’M are the derivatives of CD, CL and CM, and u(t) and w(t) denote the turbulent wind speeds along the mean wind direction and orthogonal to the mean wind direction. The definition of the wind axes is presented in Figure 1.
Figure 1. Definition of wind axes.
Since buffeting analyses are usually carried out in the frequency domain, power spectral density functions (PSDF) of u(t) and w(t) can be adopted from the literature to characterize them [7]. In this study, the PSDF given by Kaimal [8] is adopted, and defined respectively for the longitudinal fluctuating wind speed u and the vertical fluctuating wind speed w as [7],
nS(,) z n 200 f (2a) u 2 5/3 uf* (1 50 )
nS(,) z n 3.36 f w 2 5/3 (2b) uf* (1 10 ) where n is the frequency of wind fluctuations in Hz; f is the Monin coordinate defined as f = nz/U; u* is the friction velocity. To account for spanwise correlation, the cross-spectral density function of longitudinal fluctuation u at two points si and sj, Suu(si, sj, n) is employed and expressed as [7], ˆ (3) Suu(si , s j ,n) S u(z,n)exp( fu )
ˆ ˆ where fu and f w are frequency depend functions defined as,
n (4a) fˆ C x x u U x i j
n (4b) fˆ C z z w U w i j where Cx is the exponential decay coefficient for the spanwise coherence of along wind fluctuation; Cw is the exponential decay coefficient for the coherence of vertical wind fluctuation; xi and xj represent the coordinates along the span of the two given points. Since no information is available on the cross correlation between u and w, this is not considered in the analyses.
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3 DESCRIPTION OF THE BRIDGE The Baluarte bridge is located between the states of Sinaloa and Durango, in the North of Mexico, 50 km away from the Pacific coast. Figure 2 shows a picture of the Baluarte bridge, which is the tallest cable-stayed bridge in Mexico. The total length of the cable-stayed bridge is 1124 m, with a main span of 520 m and two lateral spans of 250 and 354 m, respectively. The bridge is supported by 8 reinforced concrete frame piers, two pylons with diamond shape, and two abutments at the ends. The height of the piers ranges from 40 to 140 m, approximately. The tallest pylon has a height of 165 m, while the other has a height of 147 m. The bridge connects a highway through a canyon of about 390 m of depth. A total of 76 stays are used with a semifan layout in two planes to support the bridge deck.
Figure 2. The Baluarte bridge.
It is observed in Figure 2 the intricate topography of the site of construction of the bridge, this type of terrain affects the flow approaching the bridge, and should be considered in the analyses.
The main span has a composite section with an approximate deck width of 19.80 m, the lateral spans have prestressed concrete box dowels with 22 m of deck width, approximately. An elevation view of the bridge and typical cross sections for the main and lateral spans of the bridge are shown in Figure 3.
Figure 3. Elevation view and cross sections of the Baluarte bridge.
An aerodynamic improvement to the cross section of the main span of the bridge was proposed after wind tunnel test of a section model [9]. After the tests, it was suggested to include deviators at the lateral sides of the cross section of the main span. Baffles underneath the cross section of the main span were also suggested. The lateral deviators are also shown in Figure 2.
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4 DESCRIPTION OF THE MATHEMATICAL MODEL OF THE BRIDGE
A mathematical model of the cable-stayed bridge was developed with Ansys Parametric Design Lenguage (APDL) [10]. The code to develop the geometry of the bridge was written in the note pad. The advantage of using APDL is the flexibility to easily change the parameters of the model and to carry out systematic analysis without using the graphic interface. The mathematical model developed consisted of frame elements and link elements to represent the cables. Figure 4 shows some views of the mathematical model.
Figure 4. Views of the mathematical model.
The steps followed to develop the mathematical model were:
. To draw the spine structure with a CAD program . To import the geometry from the CAD program to APDL ANSYS . To characterize each element of the model with the properties from the executive project . To define the type of analyses to carry out (Static, Modal and Harmonic)
Rigid elements were attached to the spine elements of the deck to connect all the cables. Each cable was characterized with the corresponding structural steel area and tension.
5 ANALYSIS AND RESULTS
Three types of analyses were carried out: modal, static and harmonic analyses. The modal analysis was carried out to identify the modes and frequencies associated with lateral, vertical and torsional vibration. The following table presents the first five mode shapes and frequencies extracted from ANSYS.
Table 1. Description of the first five modes of the bridge. Mode No. Mode shape Frequency [Hz] 1 Vertical 0.235 2 Lateral 0.255 3 Longitudinal 0.315 4 Longitudinal + vertical 0.442 5 Torsion + Longitudinal + lateral 0.450
It is observed in Table 1 that the first modal shape of the bridge is a vertical flexion mode with a natural frequency equal to 0.235 Hz, the second modal shape is a lateral flexion mode with natural frequency equal to 0.255 Hz, this frequency is close to that of the first mode of vibration. The third modal shape is a longitudinal mode of vibration with frequency equal to 0.315, while the fourth modal shape is a combined mode with longitudinal and vertical flexion and frequency equal to 0.442 Hz. The fifth mode is a combined mode with predominant torsional motion and natural frequency equal to 0.45 Hz. The following figure shows the first vertical, lateral and torsional modes.
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Figure 5. Vertical, lateral and torsional modes of the bridge.
The buffeting analysis was carried out with a procedure implemented in Ansys by Hu [11] for a bridge modeled as a simply supported beam and modified by Hérnandez-Landero [12] for the complete cable-stayed bridge. For all the analysis presented in this work, a mean wind speed equal to 36.24 m/s was employed, this speed corresponds to the design mean wind speed of the bridge, which corresponds to a return period equal to 200 years, and an angle of attack equal to zero. The mean (static) part of the buffeting forces were calculated and applied to the mathematical model as shown in the following figure. It is noted in the figure that the static forces were applied to the deck, pylons and cables.
Figure 5. Mean buffeting forces applied to the mathematical model (the load applied to the cables is not shown for a better visualization).
Figure 6 compares the mean horizontal and vertical displacements induced by the buffeting forces at the midspan of the deck. It is observed that the horizontal and vertical displacements are maximum at midspan, as expected, and that the former is greater than the later. It is also observed that the mean displacement at the side spans of the bridge is practically negligible, this is due to the large lateral stiffness provided by the frame piers to the structure.
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10 Lateral (p) Mid span Vertical (h) 8
6
4
Mean Displacement (cm) 2
0 1 50 100 150 200 250 300 350 400 450 487 No. Element Figure 6. Comparison of mean horizontal and vertical displacements at the midspan of the deck.
The peak response induced by the buffeting forces was obtained with a harmonic analyses. A damping ratio equal to 0.5 % for all the modes of vibrations was adopted for the analyses. Two cases were studied, the first one considers full correlation between the longitudinal forces, while the second one considers partial correlation among them. To characterize the buffeting forces, the aerodynamic coefficients for the deck section with deviators and baffles were adopted from wind tunnel test [9]. It is important to point out that for this study, the aerodynamic derivatives were not considered in the analysis; these are included in an ongoing study. The following figure presents a comparison of the PSDF of the longitudinal and vertical displacements at midspan of the deck by considering fully and partially correlated buffeting forces.
0 10 Full correlation a)
/Hz) Partial correlation
2
-5 10
PSDF displacement PSDF (cm -10 10 0.001 0.01 0.1 0.5 f (Hz)
1 10 Full correlation b)
/Hz) Partial correlation
2
-5 10
-10
PSDF displacement PSDF (cm 10
0.001 0.01 0.1 0.5 f (Hz) Figure 7. PSDF displacement at midspan of the deck for: a) longitudinal; b) vertical.
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It is observed in Figure 7 that there are differences between the PSDF for the fully correlated case and the partially correlated case. This is expected since partial correlation among longitudinal forces induces asynchronous vibration of the bridge deck, causing a different energy distribution along the frequencies. Another observation from Figure 7a is that in the partial correlation case, other combined modes of vibration (lateral+vertical) are also excited.
The maximum longitudinal and vertical displacements were calculated and compared with the results obtained from wind tunnel tests of a full aeroelastic model for two different exposures. Figure 8 shows the exposures considered in the test. More details about the tests and modeling can be found in [13].
(a) (b)
Figure 8. Exposures considered in the tests: (a) open terrain; (b) site conditions terrain.
The comparison of the analytical and experimental results for the lateral displacement at midspan deck is presented in Figure 9.
40 Mean 35 Peak 30
25
20
15 Displacement (cm) Displacement 10
5
0 Wind tunnel (open Wind tunnel (site Fully correlated (0.5 Partially correlated terrain) conditions terrain) % damping) (0.5 % damping) Figure 9. Comparison of analytical and experimental maximum displacements
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It is observed in the figure that the peak displacement by considering full correlation of the longitudinal component of wind is greater than that for the partial correlation case. It is also observed that the results from the partially correlated case are very similar to those from wind tunnel test for open terrain. When full correlation is considered, the peak response increases around 50 % with respect to wind tunnel tests for open terrain. The response from both cases: fully and partially correlated, is smaller than that from wind tunnel tests for site condition terrain. This can be due to that in the analytical model, the effects of the impact of the topography on the wind flow were not taken into account.
6 FINAL COMMENTS
A mathematical model of the tallest cable-stayed bridge in Mexico was developed with APDL code in order to study the response of the bridge under partial and fully correlated wind forces. A harmonic analysis was carry out to study the response of the bridge under buffeting forces. Particular conclusions are as follows:
. Spanwise correlation affects the peak response of the structure, when full correlation is considered, the peak response increases around 50 % of the response for partially correlated wind forces.
. The results from the partially correlated case are very similar to those from wind tunnel test for open terrain.
. The response from both cases: full and partial correlation, is smaller than that from wind tunnel tests for site condition terrain. This can be due to that in the analytical model, the effects of the impact of the topography on the wind flow were not taken into account.
ACKNOWLEDGMENTS The financial supports of the Instituto de Ingeniería de la Universidad Nacional Autónoma de México through the International Project ‘Determinación de velocidades críticas para el inicio de inestabilidad de puentes flexibles antes los efectos del viento’ and the University of Western Ontario from Canada are gratefully acknowledged.
REFERENCES [1] D.K. Sun, Y.L. Xu and J.M. Ko, Fully Coupled Buffeting Analysis of Long-Span Cable-Supported Bridges: Formulation, Journal of Sound and Vibration, 228(3), 569-588, 1999. [2] A. G., Davenport and J.P.C. King, ‘The incorporation of dynamic wind loads into the design specification for long span bridges, ACE Fall Convention and Structures Congress, New Orleans, Louisiana, 1982. [3] E. Simiu and T. Miyata, Design of Buildings and Bridges for Wind: A Practical Guide for ASCE-7 Standard Users and Designers of Special Structures, John Wiley & Sons, Inc., Hoboken, New Jersey, 2006. [4] A. G. Davenport, Buffeting of a Suspension Bridge by Storm Winds, J. Struct. Div., ASCE 3688.ST5 "Discussion", Oct. 1963, pp. 255-260. [5] The University of Western Ontario, Wind Tunnel Testing: A general outline, Boundary Layer Wind Tunnel Laboratory, May, 1999. [6] A.G. Davenport, The dependence of wind load upon meteorological parameters, Proceedings of the International Research Seminar on Wind Effects on Buildings and Structures, University of Toronto, Toronto, pp. 19-88, 1968. [7] E. Simiu and R.H. Scanlan, Wind effects on structures, 2nd Edition, John Wiley and Sons, Inc., Nueva York, 1986. [8] J.K. Kaimal and J.J. Finnigan, Atmospheric boundary layer flows, New York: Oxford University Press. 289pp, 1994. [9] O. Flamand. Wind Tunnel Tests on the Baluarte Bridge. CSTB, Nantes, 2003. [10] ANSYS Release 13.00 (Academic version), 2010. [11] Z. Hu, Gust Response of bridges to spatially varying wind excitations and calibration of wind load factors, University of Western Ontario, London, Ontario, 2009. [12] H. Hernández-Landero, Análisis de un puente flexible ante los efectos de viento turbulento, Universidad Nacional Autónoma de México, Tesis de Maestría, 2015. (In Spanish). [13] R. Gómez and A. Pozos, Reporte de pruebas de túnel de viento del puente Baluarte, 2011. (In Spanish).
14th International Conference on Wind Engineering – Porto Alegre, Brazil – June 21-26, 2015