EFFECT OF WIND DRAG ON FLUTTER OF LONG-SPAN GABLE-STAYED BRIDGES

Abstract

Cable-stayed bridges are susceptible to wind induced oscillations as they are relatively light weight, flexible and lightly damped. This flexibility combined with low damping is fundamentally responsible for the various types of aeroelastic oscillations. Unlike in the more conventional and comparatively rigid structures, the elastic deformation plays an important role in determining the external loading itself. However, the magnitude ofthe aerodynamic force is not known until the elastic deformation ofthe bridge deck is determined. This aeroelastic or self-excited phenomenon grows and the aerodynamic force may increase rapidly at specific wind speeds with the result that the bridge structure may become unstable (flutter) and may even collapse. Substantial developments in the field ofaerodynamics, after the collapse ofTacoma Narrows Bridge in which flutter was considered the main culprit, have helped planning of super long cable stayed bridges of main spans 1088 min China (Sutong Bridge) and 1018 min Hong Kong (Stonecutters Bridge) after the successful commissioning ofNormandy Bridge in France (856m) andTatara Bridge (890 m) in Japan. This thesis addresses the problem of flutter in a super long-span cable-stayed bridge with particular emphasis on self-excited drag forces which play a significant role in expediting the onset of flutter instability, and the overall flutter stability scenario based on full set of experimentally determined flutter derivatives. This aspect of study was identified after a detailed literature survey wherein self-excited drag force and lateral motion have been ignored in tradition with the airfoil flutter, or with the notion that they have little or no effect, or have been given second hand treatment with the incorporation of semi empirical lateral flutter derivatives. It is considered worth to examine the issues involved in the stability analysis ofa super long-span cable-stayed bridge. Cable stayed bridges ofmore than 1000 mspan are expected in the coming years, though they do not exist at present. Such a bridge would need a comprehensive aerodynamic analysis. Therefore, for the present study, a super long-span cable stayed bridge of 1020 mcentral span and two equal side spans each 375 mlong with biplaner cable arrangement was undertaken for aerodynamic study. Firstly, the bridge had to be designed since a cable-stayed bridge in this span range does not exist yet. The design carried out is based on a steel deck of25.5 mwidth and maximum depth of2.35 min the center. The bridge has inverted Yshaped concrete pylons of 223.6 mheight above the in deck. The design considers all significant nonlinearities and is based on the trends of existing modern long span bridges. Nonlinear analysis of the 3D finite element discretised model of the bridge was carried out with STAAD.Pro 2001/2004 software with a parallel development ofMATLAB programme EIGEN. Undamped free vibration analysis of the 3D finite element discretised model of the bridge has been carried out by the programme EIGEN developed for the purpose. Guyan condensation of the degrees of freedom was carried out to two different orders to assess the accuracy in the extent of reduction of degrees of freedom. The MATLAB Programme EIGEN was validatd with the STAAD.Pro software. Wind tunnel tests on a spring suspended section-model of the bridge deck were performed in smooth flow in a 2.1 m x 2.0 m industrial wind tunnel and the motion responses measured to extract all the 18 flutter derivatives. A thin plate section-model was also tested. Covariance driven stochastic subspace identification technique which utilizes only the output response data was used for simultaneous extraction of all the flutter derivatives using a MATLAB programme COVSSI to extract the same. The flutter derivatives were also derived from Theodorsen's Function and Quasi-steady theory for the purpose of comparison of the flutter speed obtained. Rational function approximation of the flutter derivatives was carried out to express them as a continuous function of reduced velocity and model the unsteady and nonlinear self-excited forces. The optimum parameters of the rational functions were obtained by nonlinear least-squares fit of experimentally obtained flutter derivatives by employing Levenberg-Marquardt method. A FORTRAN programme HPA was written for this purpose. Static aerodynamic force coefficients were also determined from wind tunnel test on rigidly supported section model. The results obtained have been presented and their implication discussed. A 3D multimode flutter analysis of the finite element discretised cable-stayed bridge was performed by using all the 18 flutter derivatives in expressions for self-excited lift, drag and moment forces. The equation of motion was expressed in frequency independent modal state-space form which resulted in aeroelastically modified state matrix which is a function of wind speed only. The MATLAB programme EIGEN was extended to tackle the aeroelastic analysis and to determine the eigenvalues of the unsymmetric aeroelastically modified state matrix. The logarithmic damping and frequencies obtained from the complex eigensolution of the state matrix were used to iv establish the critical flutter speed. Lateral flutter derivatives, individually and in combinations, were used to identify their roles in flutter phenomena. Flutter speeds were also determined from semi-empirical methods and compared. The bridge response to the simultaneously acting aerostatic lift, drag and moment was also determined. The main findings are summarized as follows: (i) The Particle mass matrix should not be used for mass modeling especially in long span cable-stayed bridge, (ii) The first mode for the bridge which dominates the aerodynamic (flutter) response is the lateral (transverse) mode (which may be expected of such a slender bridge). In the higher modes, the frequencies become closely spaced and mode shapes become complicated, (iii) Covariance driven stochastic subspace method performs satisfactorily in simultaneous identification of most of the flutter derivatives from the output-only measurements, (iv) Almost all the experimental flutter derivatives are differing from those determined from Theodorsen's and Quasi-steady theory both in magnitude and trend (barring Hi*, H3*, At,* and A6*). P\* which is more commonly adopted from Quasi-steady theory differs much. Unlike the Quasi-steady P4* and P6*, experimental values are not negligible, (v) The importance oflateral modes is significant. The first lateral symmetric mode is seen to participate actively in the coupled flutter phenomena where the next two modes show strong coupling, (vi) No greater role of torsional modes is seen in the critical flutter wind speed established. However, they are seen to coalesce with adjacent mode at a higher wind velocity, (vii) The role of lateral flutter derivatives has been identified. P2* and Pi* have been found to contribute little; P4*, both in isolation as well as in combination with Pi*, is the largest contributing flutter derivative. Pi* and TV contribute moderately, (viii) Exclusion of all the lateral flutter derivatives overestimates the flutter onset wind speed by 14% while the modes participating in the flutter phenomenon remain more or less the same. (ix) Exclusion of H5*, H6*, P5*, P6*, A5*, A6* also slightly escalates (about 4%) the flutter onset wind speed which is marked by zero damping and coupling of first lateral symmetric and first vertical symmetric modes. (x) The Den Hartog criteria should not be used for surmising the aeroelastic behaviour of a long span cable-stayed bridge deck. (xi) The critical flutter speeds obtained from Selberg's Formula as well as those obtained by the use of Theodorsen's and Quasi-steady theories are much higher compared to the critical flutter speed of the bridge obtained by the application of all 18 experimentally measured flutter derivatives. Therefore, for accurate estimation of the critical flutter speed for long and super long span bridge decks, multimode flutter analysis with unsteady aerodynamic forces in lift, drag and moment directions employing full set of experimentally measured flutter derivatives on a 3D nonlinear model should be carried out. vi