Usually, high-power laser beams are amplitude modulated and have phase fluctuations (PFs). The analytic formula of the effective radius R of curvature of truncated laser beams with amplitude modulations (AMs) and PFs propagating through atmospheric turbulence is derived by using the statistical optics method. It is shown that the R in free space increases with the increases of the phase fluctuation parameter, the intensity modulation parameter and the truncated parameter. And the influence of turbulence on the R also increases with the increases of the three parameters. Furthermore, the R of Gaussian beams is largest in free space, but it is most affected by turbulence. Therefore, in turbulence the R of truncated laser beams with AMs and PFs is even larger than that of Gaussian beams when the propagation distance is large enough. In particular, the relative effective radius Rr of curvature varies non-monotonically with propagation distance, and there exists a minimum at a propagation distance where the influence of turbulence on the R is largest. In addition, the position where the Rr reaches its minimum increases with the increasees of intensity modulation and phase fluctuation.