EACWE 5 Florence, Italy 19 th 23 rd July 2009 Flying Sphere image Museo Ideale L. Da Vinci Aeroelastic Analysis of Miradoiros Bridge in La Coruña (Spain) J. A. Jurado, S. Hernández, A. Baldomir, F. Nieto, School of Civil Engineering of the University of La Coruña. jjurado@udc.es Keywords: Cable stayed bridge, Aeroelasticity, flutter, sectional model, flutter derivatives. ABSTRACT In this paper the task of obtaining flutter derivatives and the computer evaluation of flutter speed was carried out for a cable stayed bridge with a main span of 658 m to be designed over the Ría of La Coruña in Spain. The bridge will be located in a very windy site and therefore some studies are required to find out the safety level against flutter aeroelastic phenomena. The set of eighteen flutter derivatives is important information required to carry out aeroelastic analysis aimed to identify the safety level of cable supported bridges against wind induced phenomena. In the case of flutter derivatives are the components of the aeroelastic damping and stiffness matrices that relate the lift, drag and moment forces to the vector of displacements and velocities of bridge deck. So far flutter derivatives are usually obtained by testing a reduced model of a deck segment in a wind tunnel. For doing that, two different alternatives exist: forced vibration approach or free vibration approach. In this occasion the free vibration approach with the sectional model of the deck supported by eight vertical and four horizontal springs was used and by carrying out tests for different speed flow. The complete set of eighteen coefficients A * i, P * i, H * i (i = 1, 6) was identified. Once flutter derivatives are obtained, flutter speed of the complete bridge can be calculated as it was explained in Jurado and Hernández (2000), by the computational solving of a non linear eigen-problem which comes from the dynamic equilibrium equation for the deck. Contact person: Dr. José A. Jurado, Associate Professor, Caminos Campus Elvina 15071 La Coruna (Spain), Tf: 34981167000, Fax: 34981167170, E-mail :jjurado@udc.es
LOCATION AND ESTRUCTURAL FEATURES OF THE BRIDGE Miradoiros Bridge has been designed as a solution for the traffic congestion problem generated as the existing bridge over the Ría of La Coruña (Spain) is not able to cope with the current number of vehicles. The figure 1 shows its location. The proposed alternative is a new cable stayed bridge upstream the existing bridge. Figure 1: Location of future Miradoiros Bridge. The projected bridge has a main span of 658 m and two secondary spans of 270 m each. The chosen deck cross-section is a 34 m wide and 3 m depth symmetric aerodynamic box girder. Special care has been taken in the aesthetics of the bridge. One of the key issues of this proposal is the envision of the bridge receiving pedestrians as it will communicate two populated urban areas and the bridge surroundings will attract a number of visitors due to the wonderful existing landscape. Its balcony zones have been dedicated around the towers at deck level to allow the recreational use of the structure. In fact, Miradoiros means balcony with beautiful views in Galician language. The figures 2 to 4 show the main dimensions of the bridge and a virtual view. Figure 2: Lateral view of Miradoiros Bridge.
Figure 3: Deck cross section and towers with balcony. Figure 4: Virtual view of Miradoiros Bridge. A three-dimensional model of the structure has been developed using ABAQUS program and beam elements. In this model the bridge deck is connected to the plane of cables by mean of fictitious beams of high stiffness. Up to twenty four modes have been obtained in order to evaluate the flutter response of the structure. In figure 5 the structural model is showed and in table 1 the natural frequencies of the mode shapes are presented along with the type of displacements associated. Only the modes which have movements for the deck are considered in the flutter analysis.
Figure 5: Structural model of Miradoiros Bridge. Tabla 1: Mode shapes and natural frequencies of Miradoiros Bridge. FLUTTER DERIVATIVES DETERMINATION To avoid experimental tests of completed bridge models in large wind tunnels that are complicated and expensive, it is necessary to use a hybrid method which is computational based but needs experimental parameters. Sectional models of the deck are initially tested in an aerodynamic wind tunnel of smaller dimensions to obtain the flutter derivatives. These coefficients are then used in the computational analysis of the aeroelastic behaviour of the completed bridge. Figure 6 shows the three forces acting on a deck. According to Simiu and Scanlan (1996) formulation, these actions are linealized as functions of the displacements and velocities of the system for vertical w, lateral v and torsional rotation ϕ x degrees of freedom. The expressions can be written as.
* * * * * * Da P1 P5 BP 2 v& P4 P6 BP 3 v 1 2 * * * 1 2 2 * * * fa = La = ρu KB H5 H1 BH2 w ρu K H6 H4 BH3 w 2 & + * * 2 * 2 * * 2 * M a BA5 BA1 B A 2 ϕ x BA6 BA4 B A 3 ϕ & x f = C u& + K u a a a where B is the deck width, ρ is the air density, U is the mean wind speed, K = Bω/U is the reduced frequency with ω the frequency of the response, and P * i(k), H * i(k), A * i(k) i = 1...6 are the flutter derivatives which are functions of K. K a and C a are called aeroelastic matrices. (1) Figure 6: Aeroelastic wind forces and displacements of a sectional model. Sectional test of Miradoiros Bridge. Sectional tests of bridge desks are carried out in aerodynamic wind tunnels. One meter square of working section is enough. The model must have the same shape of the real prototype considering barriers, aerodynamic appendages and any element which can affect to aerodynamic responses. Typical geometric scales for this tunnel size are 1/50 to 1/150. It is recommendable a model three times longer than wider, so a deck with great width requires smaller scales. As figure 7 shows, the support system is set up by means of vertical and horizontal springs which permit the three considered degrees of freedom v, w, ϕ x. The frequency similarity is not necessary to evaluate flutter derivatives because they are functions of the reduced velocity U* = 2πU / ωb. Changing the wind speed in the tunnel and the stiffness constants of the springs, a wide range of reduced velocities can be simulated. A more detailed description of sectional tests can be found in Nieto, Jurado and Hernández (2005). The dynamic equilibrium equation for the sectional model is ( C C ) u + ( K K ) u 0 M u& & =, (2) + a a where u = (v,w,ϕ x ) T. Multiplying by M -1 and denoting C m = M -1 (C-C a ) and K m = M -1 (K-K a ) becomes u & C u + K u = 0. (3) + m & m To obtain the flutter derivatives, all terms of C m and K m matrices are evaluated from the time history of the model displacements at free vibration. It is necessary to identify the natural frequency ω and the damping ratio ξ of each degree of freedom. From them, the terms of K m and C m are calculated. Denoting K U ij and C U ij the terms for a wind speed U in the tunnel and denoting K 0 ij and C 0 ij the terms for zero speed U = 0, which correspond to null aeroelastic matrices (K a = C a = 0), any flutter derivative can be evaluated by subtraction. For example, A * 2. is obtained from 0 ( C C ) 22 1 2 2 Bω B = ρ U B A2. (4) 2I U U U * 22
where I is the polar inertia of the deck which appears in the mass matrix. From (4) 2I 0 U ( K ) = ( C C ) * A2 4 22 22 ρb ω. (5) K3 K4 K3.1 K4.1 K1.1 K1.2 K1.1 K1.2 K5 K6 K3.2 K5.1 K6.1 K2.1 K2.2 K2.1 K2.2 K5.2 K6.2 Figure 7: Spring support system of a sectional model deck in the University of La Coruña wind tunnel. The model is elastically sustained using eight or twelve springs: four or eight vertical and four horizontal ones. The stiffness of the springs determines the vibration frequencies (2πf = ω) of the system that together with the wind velocity in the tunnel, U and the model width, B, determines the range of reduced velocities, U*, in order to be able to obtain flutter functions. The sectional model of the Miradoiros Bridge appears on the figure 8 and its properties are summarized in the table 1. The testing was carried out for wind velocities between 3 and 11 m/s. The minimum and maximum reduced velocities with which flutter functions can be obtained are shown in table 2. Tabla 2: Properties and natural frequencies of the sectional model of Miradoiros Bridge.
Figure 8: Sectional model of Miradoiros Bridge. Flutter derivatives obtained by software PCTUVI PCTUVI is the control software for the wind tunnel of the School of Civil Engineering of the University of La Coruña (Spain). The structure of the program consists on two parts. First one is the data acquisition and uses Microsoft Visual Basic programming language. The second part is the data analysis which is programmed on MATLAB language for technical computing. The computer program can control two kind of possible tests: aerodynamic or aeroelastic. The aerodynamic test measures the lift, drag and moment that wind action causes on the fix sectional model. Curves for the aerodynamic coefficients C L, C D and C M are obtained as a funtion of the angle of attack. The most interesting achievement of PCTUVI is the simultaneous calculus of all the flutter derivatives. Figure 9: Widow for selecting the data interval of free vibration response. The user defines the values the properties of the model, the stiffness and distances among the springs, the sampling frequency and finally the atmospheric parameters. The following step is to
evaluate the equilibrium position of the model as reference of the coordinates. Then, the displacements of free vibration are registered (figure 9) without speed of wind in the tunnel. Afterward a serial of tests increasing the wind speed in the tunnel are carried out. One of the most efficient methods to obtain the frequency and damping properties of a free vibration system was proposed by Sarkar, Jones and Scanlan (1992) and it is based on Ibrahim and Mikulcik time domain method (1977). The frequency ω and the damping ξ for each degree of freedom of the sectional model is calculated from an eigenvalues problem in which the matrix is built with the time history of the sectional model. From this vibration parameters, it is possible to obtain the stiffness K m and damping C m matrices of the system. Each test corresponds to one point on the curves of the flutter derivatives. When the speed of wind is not zero, the obtained flutter derivatives are saved. Figures 10 to 12 show the obtained flutter functions for Miradoiros Bridge. Figure 10: Flutter derivatives related with the aeroelastic moment.
Figure 11: Flutter derivatives related with the aeroelastic lift.
Figure 12: Flutter derivatives related with the aeroelastic drag. The results of flutter derivatives are similar for attack angle of 0º and -1.5º. However, the A * 1, A * 2, A * 4, H * 2, and P * 3 have little differences for the positive angle of attack 1.5º. FLUTTER ANALYSIS OF THE MIRADOIROS BRIDGE In the present work, a coherent matrix formulation has been used for the computational phase of hybrid flutter analysis. Jurado and Hernandez (2004) explain this formulation stems from the equation (2) ensambling the matrices and vector for the full structural model. Through modal analysis it is possible to approximate the deck displacements by means of a linear combination of the most
significant mode shapes. Assembling them in columns into the modal matrix Φ, the displacement vector can be expressed as u = Φq. Each element of the vector q represents the participation of each mode shape in the displacement vector u. Premultiplying (2) by Φ T it becomes Iq&& + C q& + K q = 0 (6) R R where C R = Φ T (C - C a ) Φ, K R = Φ T T (K - K a) Φ and Φ MΦ = I Using mass normalized modes. Knowing that the solution of this equation has the form q(t) = we µt, becomes ( µ µ ) Iw + C w + K w = 0 (7) 2 t R R e µ which can be transformed into an eigenvalue problem by adding the identity µiw+µiw = 0: I 0 µ w C K µ w µ + 0 I w I 0 w or in short ( ) t R R t e µ = 0 A µ I w e µ =0 (9) µ The imaginary part β of the eigenvalues µ counts on the frequency ω, while the real part α of the eigenvalues is associated with the damping ratio ξ. The condition of flutter corresponds to the lowest wind speed U f which gives one eigenvalue with vanished real part. However, the problem (9) is non-linear because the matrix A assembles the aeroelastic matrices K a and C a. These matrices contain the flutter derivatives, which are functions of the reduced frequency K = Bω/Ū, and the frequency for each eigenvalue ω remains unknown until the problem has been solved. Figure 13 shows graphs with the evolution of the eigenvalues of the problem for increasing wind velocities. It is observed that the critical flutter velocity for 0º of attack angle and using 20 mode shapes is 98.11 m/s. Others analyses are shown in table 3. (8) Figure 13: Graphs with the evolution of the eigenvalues for increasing wind velocities and critical flutter speed and reduce frequency.
Tabla 3: Results of flutter analyses for Miradoiros Bridge. CONCLUSIONS A hybrid method with experimental and computational phases facilitates the calculation of flutter speed since it is cheaper than an entire bridge model testing in a large wind tunnel. PCTUVI software has been used efficiently for aeroelastic experiments with sectional model bridges. This program uses the Modified Ibrahim Time Domine method (MITD) to analyse the free vibration response of the model and from its displacements it is possible to obtain the 18 flutter derivatives at the same time. For Miradoiros Bridge, a cable stayed bridge with a main span of 653 m, the results of critical flutter speed considering 20 mode shapes are very similar to the analyses which use the three modes with more participation. The extra flutter derivatives calculated with the lateral degree of freedom in the wind tunnel have not special influence in the flutter results. Bridges with a main span of more than one kilometre long present much more differences when 18 flutter derivatives are considered. Beside, for those cases, a multimodal analysis with a great number of mode shapes is crucial. Increasing the number of modes and flutter derivatives considered in the flutter analysis does not represent a significant increment in the associated testing or computational costs. Working with a high number of modes and the complete set of flutter derivatives means higher accuracy in the evaluation of the flutter speed. ACKNOWLEDGMENTS This work has been funded by the Spanish Ministry of Science and Innovation under project BIA2007-66998. REFERENCES Jurado J. A. and Hernández S. (2000). Theories of Aerodynamic Forces on Decks of Long Span Bridges. J. of Bridge Eng. 5, n1 8-13. Simiu E., Scanlan R. (1996). Wind effects on Structures. John Wiley & Sons, New York Jurado J. Á., León A., Hernández S. (2005) Wind Tunnel Control Software for Identification of Flutter Derivatives on Bridge Sectional Tests. EACWE 4. 4º European and African Congress in Wind Eng. Prague, Check Republic. Sarkar P. P., Jones N. P. y Scanlan R. H. (1992) System identification for estimation of flutter derivatives, Journal of Wind Engineering and Industrial Aerodynamics 41-44 1243-1254. Ibrahim S.R. y Mikulcik E. C. (1977) A Method for the Direct Identification of Vibration Parameters from the Free Response, The Shock and Vibration Bulletin, bulletin 47, Part 4. Jurado J. Á., Nieto F., Hernández S. and Mosquera A. (2007) Efficient cable arrangement in cable stayed bridges based on sensitivity analysis of aeroelastic behavior. Advances in Engineerin Software, doi:10.1016/j.advengsoft.2007.10.004. Jurado, J. Á., Hernandez S. (2004) Sensitivity analysis of bridge flutter with respect to mechanical parameters of the deck. Structural and Multidisciplinary Optimization Vol. 27, Nº 4. June, 2004.