Dynamic Buckling of Functionally Graded Timoshenko Beam under Axial Load

In this study, we investigated the dynamic buckling of the functionally graded Timoshenko beam whose property parameters continuously change according to the power function along the thickness direction.Based on the first order shear deformation theory, we derived the governing equation of the dynamic buckling of functionally graded material Timoshenko beams under axial step loading by using the Hamilton's principle.Using the Ritz method combining with the de Moivre's formula, we obtained the buckling solution and the expression of the critical load of the dynamic buckling of functionally graded material Timoshenko beam under the clamped-fixed boundary condition.Then, the influence of geometric size, gradient index, modal number, material composition, Poisson's ratio and elastic modulus on the critical load by MATLAB calculation was discussed.The results show that the critical load of the functionally graded material Timoshenko beam decreases with the increase of beam length and the gradient index, and increases with the increase of the modal number, showing that the higher modal number is more easily excited by the increase of impact load.Furthermore, the critical load increases with the increase of the Poisson's ratio and the elastic modulus, and the effect of elastic modulus is greater than Poisson's ratio.The critical load-critical length curve tends to be gentle at the loading end because of the influence of shear term.Buckling mode of beam becomes more complicated when the modal number increases.