Aerodynamic Stability of Long-span Suspension Bridges
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A MATLAB script for making customized finite element models of suspension bridges is created. The main purpose of the script is to perform eigenvalue analysis of different bridge setups to investigate the controlling parameters for the torsional-to-vertical frequency ratios. The modal output from the model is used to perform aerodynamic stability analysis. The in-wind complex eigenvalue problem (CEV) is solved in an iterative procedure in order to locate the stability limits of the various bridge setups. The bridge geometries investigated originates from a conceptual study on triple-girder flutter-free bridges conducted by Michael Styrk Andersen at The University of Southern Denmark. The FE-model is verified by analysing the Hardanger Bridge. Both the model output and the instability limit, found as 77.5 m/s, is in good agreement with previous research. The results for other setups gave reasonable natural frequencies and mode shapes, and are verified by simplified calculations made by Michael Styrk Andersen. The widest triple-girder configuration, Setup 3, shows below unity frequency ratios, as expected. The importance of pylon stiffness is investigated, and it is concluded that the natural frequencies are not very sensitive to changes in pylon stiffness. An analysis is made to determine what effect the crossbeam stiffness has on the natural frequencies. For very stiff crossbeams the frequency ratio of Setup 3 was inverted. This allows classical flutter, reducing the stability limit considerably. The available literature has been searched for applicable aerodynamic derivatives for the configurations that are studied. The Messina ADs are implemented for the triple-girder setups. The results are not satisfactory as they are highly unstable regarding the choice of curve fitting. Therefore all setups are analysed using Hardanger and Theodorsen ADs. Michael has planned and performed wind tunnel tests on Setup 1-3 parallel with the work on this project. The data is not yet processed as this report is in its finishing stages. It is therefore left for further work to analyse the stability of these setups with the correct ADs. Besides the verification by Hardanger Bridge, there are mainly three different bridge setups analysed. They are all fictitious designs of a suspension bridge crossing Halsafjorden on the Norwegian coastal highway E39. The bridges has main spans of 2050 meters. Setup 1 is a single hollow-box girder similar to the Hardanger bridge girder. The stability limit is 26.5 m/s, and denotes the wind velocity at which the bridge deck enters coupled flutter in the first pair of symmetric modes. Setup 2 has a medium wide triple-girder bridge deck. With Hardanger ADs it undergoes classical flutter at 28.2 m/s, in good correspondence with observed behaviour in wind tunnel testing. The stability limit of Setup 3 is not identified by the complex eigenvalue procedure applied because of the low frequency ratios. The results indicate a critical wind speed of circa 71 m/s for torsional divergence. Unlike the Hardanger analysis, all setups with span-length 2050 meters has considerable lateral deflections in the anti-symmetric torsional modes. This increases the effect of the lateral ADs on the critical wind speed and frequencies. It is however observed that neglecting these ADs is conservative for all analyses in this report. The in-wind characteristics of suspension bridges with low torsional-to-vertical frequency ratios are investigated. The present results indicates that such bridges has good aerodynamic performance and that flutter instability is avoided. Achieving such low frequency ratios is though compromising for the torsional stiffness of the bridge deck girder, causing fairly low stability limits for static divergence. A CEV analysis of Setup 3 with improved torsional stiffness is conducted, indicating an improvement of the wind velocity at which static divergence occurs. The results indicate a critical wind velocity for static divergence of 91 m/s when increasing the cable distance from 30 to 40 meters. This result verifies indications made in existing research on similar bridges.