Harmonic analysis of elliptical hollow section tubes in blending.

When tubular members are in bending, they tend to flatten towards the axis of bending, thus reducing their second moment of area. Eventually, a limit point is reached whereupon the initial stability of the system is lost and unstable equilibrium prevails, thus reducing the load carrying capacity of the member. This effect was originally described by Brazier (1927) for circular tubular members. In the present study, the analysis of Brazier is adapted for elliptical hollow section members, taking into account the additional geometric complexities inherent in ellipses. An analytical method is presented whereby the initial geometry and the displacement functions of the system are replaced by Fourier series, thus reducing the analytical complexity of the problem. After formulating the potential energy functional, use of a variational method allows for the amplitudes of the constituent harmonics of the Fourier approximations of the displacement functions to be solved for, providing estimates of the deformed geometry of the cross-section and the associated moment. In keeping with the analogy of Brazier for circular sections, a limit point is observed. These analytical predictions are then compared with the results of a complementary finite element analysis, whereupon it is found that for smaller longitudinal curvatures there is close agreement between the analytical and numerical methods. For larger curvatures and moments beyond the limit point some divergence is observed between the predictions of the two methods, which can be attributed to the lower-order approximations assumed in formulating the potential energy functional in the analytical method