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Buckling of multiwalled carbon nanotubes (MWCNTs) subjected to bending deformation is studied using molecular dynamics simulations. We show that the initial buckling mode of a thick MWCNT is quite different from that of a thin MWCNT. Only several outer layers buckle first while the rest inner layers remain stable in a very thick MWCNT, while in a relatively thin MWCNT, all individual tubes buckle simultaneously. Such a difference in the initial buckling modes results in quite different size effects on the bending behavior of MWCNTs. In particular, the critical buckling curvature of a thick MWCNT is insensitive to the tube thickness, which is in contrast with linear elasticity. It is found also that the initial buckling wavelength is weakly dependent on the thickness of the MWCNT. We demonstrate that rippling deformation does decrease the effective modulus of a bent MWCNT, as observed in experiments. Finally, we show that the interlayer van der Waals interactions have little effect on the bending behavior of a MWCNT in the linear elastic regime.