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Wind-Induced Response of a Cable-Stayed Bridge Under Partially and Fully Correlated Buffeting Forces

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Wind-Induced Response of a Cable-Stayed Bridge Under Partially and Fully Correlated Buffeting Forces 1 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 14th International Conference on Wind Engineering – Porto Alegre, Brazil – June 21-26, 2015 2 (1b) 1 2 2u(t) ' w(t) Lb (t) U BCL (CL CD ) 2 U U 1 2u(t) w(t) (1c) M (t) U 2B2 C C ' b 2 M U M 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], (2a) nSu (,) z n 200 f 2 5/3 uf* (1 50 ) nSw (,) z n 3.36 f 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. 14th International Conference on Wind Engineering – Porto Alegre, Brazil – June 21-26, 2015 3 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. 14th International Conference on Wind Engineering – Porto Alegre, Brazil – June 21-26, 2015 4 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.
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