Owing to their single atom-layer structure, actual bending stiffness of single-walled carbon nanotubes is much lower than that given by the elastic shell model if the commonly defined representative thickness is used. In this paper, it is proposed that the effective bending stiffness of single-walled nanotubes should be regarded as an independent material parameter not related to the representative thickness by the classic bending stiffness formula. Based on this concept, the modified formulas for the critical axial strain and the wavelength of axially compressed buckling are found to agree well with known data of molecular-dynamic simulations. On the other hand, in contrast to single-walled nanotubes, bending stiffness of multiwalled nanotubes is found to be well estimated by the classic bending stiffness formula when adjacent nanotubes are squeezed severely so that the induced high friction barrier prevents interlayer slips. In particular, these results offer a plausible interpretation for the wavelength of large-strain local buckling of multiwalled carbon nanotubes under bending observed by Falvo et al. [Nature (London) 389, 582 (1997)].