## Dynamic behaviour of cablesupported bridges subjected to strong natural wind

##### Abstract

This thesis discusses wind-induced dynamic response of slender cable-supported bridges. The focus has been on prediction of the flutter stability limit and the buffeting response in strong winds. The thesis consists of journal papers that are either submitted or published.
Multimode flutter has recently been shown to be the governing phenomenon for the aeroelastic stability limit of long-span cable-supported bridges. In this thesis the multimode flutter phenomenon is thoroughly studied. It is concluded that the most important indicator of possible multimode effects is the shape-wise similarity of the vertical and torsional vibration modes since flutter will not occur if the still-air vibration modes are shape-wise dissimilar. When the stability limit of a long-span bridge is assessed, the shape-wise similarity of all possible mode combinations should be evaluated first. Then the system should be grouped into uncoupled subsystems. The subsystem involving the still-air torsional vibration mode with the lowest natural frequency will most likely provide the lowest stability limit. If this subsystem consists of more than two vibration modes, multimode effects will occur. The reduction of the stability limit will be small if the shape-wise similarities of the vibration modes are not of the same order of magnitude, or if the system consists of two torsional and one vertical mode, and the torsional modes are well separated. If these conditions are not fulfilled, the flutter stability limit should be assessed using a multimode approach.
The self-excited forces are particularly important when the wind-induced dynamic response or the flutter stability limit is assessed for slender bridges. The self-excited forces can be modelled simply using quasi-steady theory. Since the quasi-steady theory is frequency independent, the model may be used in both the time and frequency domains. However, it is well known that the traditional quasi-steady theory may severely underestimate the flutter stability limit since no aerodynamic torsional damping is introduced into the model. In this thesis a novel modified quasi-steady theory is suggested. The method takes advantage of that the self-excited forces are most important at the natural frequencies of the combined structure and flow system. This implies that curves providing a frequency-independent description of the self-excited forces can be fitted to the experimental data in the important reduced-frequency range corresponding to the natural frequencies of the system. The suggested model has been applied for the Hardanger Bridge in a comprehensive case study, and it is concluded that the model provides wind-induced response and flutter stability limits of good accuracy.
Simplified methods for assessment of the flutter stability limit are still considered important in preliminary designs and when designing medium-span bridges where multimode effects will not reduce the flutter stability limit significantly. The most popular approach is still Selberg’s formula, published almost 50 years ago. Selberg’s formula provides the flutter stability limit with reasonable accuracy if the aerodynamic properties of the cross section are similar to those of a flat plate, and the vertical and torsional modes have an identical shape. In this thesis an alternative analytical approach to Selberg’s formula is suggested. The formulae presented are based on the fundamental flutter equations, and the simplified solution is developed by introducing two assumptions. (i) The quasi-steady model for the self-excited forces outlined above is introduced in the equations of motion. (ii) The critical frequency is assumed to be on the torsional branch of the solution system and can be approximated by the uncoupled system of equations. It is demonstrated that by introducing these two approximations, the complexity of the flutter equations is significantly reduced, and if the still-air damping is neglected, a closed-form solution of the flutter stability limit may be obtained. The formula presented is very similar to Selberg’s formula, but contains coefficients taking into account the actual aerodynamic properties of the cross section and the possible imperfect shape-wise similarity of the vibration modes. The formulae presented are tested for a range of hypothetical structural configurations, in addition to the properties of a few well-known bridges, considering the aerodynamic properties of two cross sections. It is concluded that the proposed formulae provide results of adequate accuracy, and that they can be regarded as engineering approximations of the critical flutter velocity.
This thesis also discusses unsteady modelling of the self-excited forces in the time domain. A comprehensive case study, where the wind-induced dynamic response of a slender suspension bridge is assessed in the time domain, is presented. Here, the selfexcited forces have been modelled, using rational functions, indicial functions, a novel modified rational function approach explained and introduced in this thesis, and a further developed modified quasi-steady theory. The quasi-steady model is a further development of the model outlined above. As explained above, in the modified quasisteady theory suggested in this thesis, the experimental results of the aerodynamic derivatives are approximated with curves providing a frequency-independent description of the self-excited forces in the important reduced-frequency range. This implies that the self-excited forces may be accurately modelled at frequencies corresponding to one horizontal, one vertical, and one torsional vibration mode. The further development presented here is to uncouple the aeroelastic system utilizing the right- and left-hand eigenvectors. This implies that the experimental results for the aerodynamic derivatives may be accurately approximated for all the natural frequencies of the aeroelastic system. It is concluded that all the unsteady models evaluated provide an adequate description of the self-excited forces, but that the unsteady models may be challenging to fit to the experimental data since the same coefficients are used in the expressions for the imaginary and real part of the transfer functions, which implies that two sets of data have to be approximated using the same coefficients. It is also seen that the quasi-steady model presented provides satisfying results. The results are in fact of higher accuracy than when some of the unsteady models are applied.
As modern bridges become longer, slenderer and lighter, the use of nonlinear methods to evaluate the dynamic response may become necessary. This implies that time domain assessment of the wind-induced dynamic response will become more important in the future. When nonlinearities are introduced into the model, it is an advantage to use the degrees of freedom of the element model directly and not use still-air vibration modes as generalized degrees of freedom. This can be done modelling the self-excited forces at distinct points along the girder, but this will imply that a huge amount of convolution integrals have to be evaluated at each time step. Another approach is suggested in this thesis. The starting point is a traditional beam element with twelve degrees of freedom, where the convolution integrals are added as aerodynamic degrees of freedom in each node. This implies that the convolution integrals do not need to be evaluated explicitly, since their value is calculated just like the response of the structure. Four different aeroelastic beam elements have been developed and tested. It is concluded that the elements provide wind-induced dynamic response and flutter stability limits that correspond very well to results predicted by the traditional multimode approach in the frequency domain.
Accurate modelling of the wind field is a crucial issue when predicting the dynamic response of long-span bridges. The wind field is most commonly modelled as a multivariate Gaussian stationary and homogeneous stochastic process, where the turbulence components are assumed independent. Since the wind field is affected by the roughness at the site, the turbulence components will become correlated, since threedimensional eddies are generated by the roughness elements, but this effect is normally neglected. In this thesis the measurements of the fluctuating wind carried out at the Sotra Bride in 1975 are reinvestigated. The cross-spectral densities of all the turbulence components have been determined using auto regressive (AR) models. It is concluded that the cross-spectral density of the u and w components may have a significant influence on the dynamic response, in particular for structures with low natural frequencies. However for the bridge considered, reasonable estimates of the windinduced dynamic response will still be obtained if the cross-spectral density of the u and w components is neglected, but the accuracy of the modelling will be improved if it is included.

##### Has parts

Oiseth, Ole; Ronnquist, Anders; Sigbjornsson, Ragnar. Simplified prediction of wind-induced response and stability limit of slender long-span suspension bridges, based on modified quasi-steady theory A case study. Journal of Wind Engineering and Industrial Aerodynamics. (ISSN 0167-6105). 98(12): 730-741, 2010. 10.1016/j.jweia.2010.06.009.Oiseth, Ole; Ronnquist, Anders; Sigbjornsson, Ragnar. Simplified prediction of wind-induced response and stability limit of slender long-span suspension bridges, based on modified quasi-steady theory. Journal of Wind Engineering and Industrial Aerodynamics. (ISSN 0167-6105). 99(8): 894-894, 2011. 10.1016/j.jweia.2011.07.001.

Oiseth, Ole; Sigbjornsson, Ragnar. An alternative analytical approach to prediction of flutter stability limits of cable supported bridges. Journal of Sound and Vibration. (ISSN 0022-460X). 330(12): 2784-2800, 2011. 10.1016/j.jsv.2010.12.026.

Oiseth, Ole; Ronnquist, Anders; Sigbjornsson, Ragnar. Time domain modeling of self-excited aerodynamic forces for cable-supported bridges. Computers & structures. (ISSN 0045-7949). 89(13-14): 1306-1322, 2011. 10.1016/j.compstruc.2011.03.017.

Øiseth, Ole; Rönnquist, Anders; Sigbjörnsson, Ragnar. Finite element formulation of the self-exited forces for time-domain assessment of wind-induced dynamic response and flutter stability limit of cable-supported bridges. . 10.1016/j.finel.2011.09.008.

Øiseth, Ole; Rönnquist, Anders; Sigbjörnsson, Ragnar. Effects of cross-spectral densities of atmospheric turbulence on the dynamic response of cable-supported bridges: A case study. .