Comparison of natural characters between buckling Timoshenko and Euler-Bernoulli beams under the axial force

Natural characters of the buckling Timoshenko beam under the axial force was investigated comparing with the Killer-Bernoulli one for the first time. Firstly, the partial-differential-integral governing equation of the Timoshenko beam with simply supported boundaries was established. The critical buckling axial force and the buckling non-trivial configuration were derived analytically. The stability of the nontrivial configuration was discussed together with the Kuler-Bernoulli beam. Meanwhile, the influence of the parameters on the buckling bifurcation was discussed. The Galerkin truncation method was applied to calculate the frequencies of the two models under their supercritical axial force. It was found that, alter buckling, the frequency of the Timoshenko beam is always larger than that of the Kuler-Bernoulli beam, for the critical axial force of the Timoshenko one is smaller. Part of the vibration energy of the Timoshenko beam was stored in the rotational deformation of the cross-section, which softens the bending stillness. This softening effect is more obvious for higher order frequencies, which makes higher order frequencies more different than the fundamental frequency while comparing with the Kuler-Bernoulli beam. Also, during the truncation, it can be found that higher order frequencies are no longer affected by the axial force in the supercritical region, except for the first order one. For beams with large slenderness ratio, the smaller the difference in the natural frequencies will be. This further illustrates the necessity of using the Kuler-Bernoulli beam for slender structures and the Timoshenko beam lor those short and thick. © 2022 Chinese Vibration Engineering Society. All rights reserved.