Mechanical buckling of single-wall, self-similar carbon nanotubes is investigated via Molecular Dynamics (MD) simulations with the second generation Tersoff-Brenner potential. The self-similar tubes are assumed to have the same aspect ratio (λ), defined as radius (R) divided by length (L). Four different types of armchair nanotubes were studied, i.e. (3, 3), (5, 5), (10, 10) and (15, 15). Buckling strain (εcr) is inferred from strain energy discontinuities while the tubes are under uniaxial compression. MD-calculated buckling strains are compared with experimental data of bulk cylindrical shells, as well as continuum shell buckling theories, including empirical formula. With the self-similarity, dependence of buckling strain on tube length may be characterized by the parameter, α= log(εcr)/log(L/t). It was found that the magnitude of α decreases as λ decreases, switching from the shell- to Euler-type buckling mode. Even within the shell buckling mode, the thickness parameter is different for different λ. It was found that, for λ=0.276, the wall thickness (t) of the nanotubes was estimated to be 0.066 nm in order for the shell theories predicting MD-calculated buckling strains. For λ=0.153, t was found to be 0.34 nm. In addition, under the self-similar condition, smaller nanotubes exhibit larger buckling strain, indicating that smaller tubes have higher buckling resistance.