Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler–Bernoulli beams

Analytical relations between the critical buckling load of a functionally graded material (FGM) Timoshenko beam and that of the corresponding homogeneous Euler–Bernoulli beam subjected to axial compressive load have been derived for clamped–clamped (C–C), simply supported–simply supported (S–S) and clamped–free (C–F) edges. However, no such relation is found for clamped–simply supported (C–S) beams. For C–S beams, the transcendental equation has been derived to find the critical buckling load for the FGM Timoshenko beam which is similar to that for a homogeneous Euler–Bernoulli beam. For the FGM beams Young’s modulus, E, and Poisson’s ratio, m, are assumed to vary through the thickness. The significance of this work is that for the C–C, S–S and C–F FGM Timoshenko beams, the critical buckling load can be easily found from that of the corresponding homogeneous Euler–Bernoulli beam and two constants whose values depend upon the through-the-thickness variations of E and m. For the C–S FGM Timoshenko beam the transcendental equation for the determination of the critical buckling load is similar to that for the corresponding homogeneous Euler–Bernoulli beam.