ANALYSIS OF WIND INDUCED OSCILLATIONS IN CABLE STAYED BRIDGES

Abstract

The analysis of cable stayed bridges to estimate the dynamic response due to wind loading is an imperative requirement in the design criteria, as they are relatively lightweight, flexible and lightly damped structures. The research in the area of bridge aerodynamics marked a beginning in this direction after the collapse of Tacoma Narrows Bridge in 1940s. The last two decades witnessed a lot of progress in wind tunnel testing techniques, analytical assessment and development of software to evaluate wind induced response. These advancements in the state-of-the-art, applied to the design, helped in the design of long span cable stayed bridges such as Normandy Bridge in France (main span 856m) and Tatara Bridge in Japan (main span 890m). The wind induced oscillatory phenomena can be grouped into limited amplitude oscillations such as buffeting, vortex excitation, etc. and divergent amplitude oscillations such as flutter and galloping. This dissertation is devoted to buffeting as well as flutter analysis of long span cable stayed bridges, as the former affects the safety and serviceability states whereas the latter may become critical in design. Some of the problems in the area of buffeting and flutter analysis of long span cable stayed bridges are identified after a detailed literature review. As these bridges are geometrically nonlinear structures, any attempt to simplify the analysis may lead to inaccurate prediction ofresponses. Also, usage of assumed modal structural damping in the aerodynamic analysis of these bridges might not reflect their actual behaviour under the action of wind. Therefore, there is a need to apply reliable analytical methods for evaluation of modal structural damping for rational wind analysis. Type of deck supports at towers and abutments such as fixed, movable, floating and elastic supports play an important role in design of cable stayed bridges. However, their influence on aerodynamic behaviour is not reported so far. Frequency domain buffeting analysis, even though computationally efficient, is not capable of handling nonlinearities in the system. The other option to conduct in aeroelastic model investigations of long span bridges, may demand larger size wind tunnels, or compel to make a compromise on modelling of wind and structure, which may lead to inaccurate prediction of responses under wind. Further, modelling of the bridge deck supports for wind tunnel investigations of aeroelastic models is not an easy task. The limitations of conventional methods can be overcome by performing buffeting analysis using time domain. With longer span lengths, the possibility of occurrence of coupled flutter, especially combined lateral and torsional modes, needs to be investigated. As innovations in the design of cable stayed bridges are still in progress, new generation cable stayed bridges with improved and elegant designs are likely to be constructed in the near future. Torealize these innovative long span bridges, there is a need to develop realistic analytical procedures to understand the complex wind induced oscillatory problems. After considering the state-of-the-art analytical procedures in bridge aerodynamics, a comprehensive approach for buffeting and flutter analysis of long span cable stayed bridges is developed. The steps include (i) nonlinear static analysis under dead load accounting for geometric nonlinearities, (ii) vibration analysis using dead load deformed geometry (iii) evaluation of energy based modal structural damping (iv) digital simulation of wind velocity field (v) time domain buffeting analysis using generated buffeting forces and (vi) flutter analysis. In this thesis, the above methodology has been used to study the effect of (i) terrain roughness, (ii) mean wind speed and (iii) bridge deck supports at towers and abutments, onthe aerodynamic response of cable stayed bridges. The steps involved are described below. Nonlinear Static and Vibration Analysis: The nonlinear static analysis of cable stayed bridges is performed after idealizing these bridges as a three-dimensional space system and the deformed geometry is obtained under dead loads and initial cable tension. After modifying the initial geometry, the vibration analysis is carried out using Lanczos iteration procedure and the frequencies and mode shapes are determined. IV Using the modal ordinates and element and geometric stiffness matrices of the deck, tower and cable elements, the modal strain energy and potential energy contribution of these components is computed. Using the energy loss factors corresponding to the materials ofthese bridge components, the total energy dissipated is computed. The modal structural damping is evaluated as ratio of total energy dissipated to potential energy. Software routine ENDAMP has been developed for evaluation of energy based modal structural damping. Digital Simulation of Wind: For the time domain buffeting analysis, the wind field along the span ofbridge is generated using the spectral representation method using the Kaimal spectrum and Panofsky-McCormick spectrum for longitudinal and vertical velocity fluctuations respectively. The method essentially consists ofrepresenting the components of random process/velocity fluctuations as sum of cosine functions with random frequencies and phase angles corresponding to the target cross-spectral density matrix. Computer software WTNGEN has been developed for generation of spatially correlated wind velocity field along the span of bridge. Time Domain Buffeting Analysis: The buffeting forces in the lateral, vertical and rotational directions required for performing time domain analysis are first generated by modified quasi-steady approach using the simulated wind fluctuations, steady state force coefficients and their derivatives as well as aerodynamic admittance functions. These forces are generated at a time step of 0.25 seconds, and a total duration of 512 seconds was chosen after a sensitivity analysis. The net modal damping required for buffeting analysis is obtained by summing up the modal structural and aerodynamic dampings. The aerodynamic damping in vertical, lateral and rotational directions are evaluated using the expressions, which matched well with aerodynamic damping measured in wind tunnel studies (Irwin, 1977). The buffeting analysis is performed by time integration ofequations ofmotion in modal co-ordinates using Wilson-8 method. Flutter Analysis: The formulation/ for equations of motion for flutter analysis are derived along with methodology for two-dimensional and three-dimensional flutter analyses. The possibility of occurrence of coupled lateral and torsional motion is examined. Numerical Analysis of Bridges: After duly validating the procedures for buffeting and flutter analysis as presented above, their applications are illustrated with numerical analyses of four cable stayed bridges - two each with three spans and the other two with five spans. The three span bridges are composite structures with steel deck and concrete A-shaped towers with total span length of 627.8m (Bridge # 1) and 1255.8m (Bridge # 2). The five span bridges included are : the existing steel bridge, the Luling Bridge in USA with a span of 836.6m (Bridge # 3), and a concrete bridge, the Yamuna Bridge with a total span of 610m (Bridge # 4), under construction in India. The nonlinear static and vibration analyses of the bridges are performed first. The modal structural damping is theoretically evaluated. The time domain buffeting analysis has been performed for (i) terrain categories TC-1 to TC-4 with surface roughness parameter 0.005m, 0.03m, 0.3m and 1.0m (ii) mean wind speed in the range of 30 to 60m/sec, at steps of lOm/sec and (iii) type of deck supports (DST-1 to DST-6). After performing the time domain buffeting analysis, the results are compared with the mean wind response. The gust response factor - ratio of peak to mean static response - has been evaluated to quantify the amplification in response due to buffeting. Based on gust response factor, the best type of deck supports suitable for aerodynamic design of cable stayed bridges has been selected. The effect of buffeting on forces in cables, deck, tower as well as on the reactions at tower base and deck supports at towers and abutments has been quantified by comparing the results of buffeting analysis with static response of the bridges under mean wind. Flutter analysis of these bridges is carried out to observe the effect of bridge vibration in higher modes, effect of angle of attack, influence of deck supports on occurrence of flutter, and, prediction of flutter due to coupling of lateral and torsional motions. The main findings of this study are summarized as follows: 1. Improvement in state-of-the-art wind analysis is achieved by adoption of theoretically evaluated modal structural damping for wind analysis, time domain approach for buffeting analysis and prediction of critical flutter speed vi for coupled lateral and torsional modes in long span cable stayed bridges. Detailed studies are carried out to observe the effect of type of deck supports on aerodynamic behaviour of cable stayed bridges. 2. The time domain approach used in this study to compute buffeting response of cable stayed bridges, serves as an alternate tool to aeroelastic wind tunnel testing which is time consuming and expensive. 3. Buffeting response increases nonlinearly with mean wind speed. The major contribution to buffeting response in vertical direction is by the first and second symmetrical vertical bending modes. The number of modes to be included for buffeting analysis of long span flexible bridges depends on the type of bridge deck supports and could be decided on the basis of vibration characteristics. Buffeting induced forces in outer cables, deck member near tower as well as the vertical reaction at deck supports at abutments or near cable anchorages are significantly high in comparison to forces induced by mean wind. 4. Nonlinear static response of cable stayed bridge is dependent on the type of deck supports. The mode type and order in which the bridge gets excited varies with type ofdeck supports. 5. Energy based evaluation of damping serves as an efficient analytical tool to assess the modal structural damping for the analysis and design of cable stayed bridges under random wind forces. 6. The effect of bridge vibration in higher modes on criteria for onset of flutter is to reduce the critical wind speed in long span cable stayed bridges. The angle of attack of wind plays an important role in flutter analysis, as the critical wind speed for occurrence of flutter significantly decreases with increase in angle of attack (positive). Therefore, it is necessary to determine the flutter derivatives at various wind angles and perform flutter analysis at different angles of attack of wind for a rational wind design. vn