By using the complex Gaussian function expansion method and the Wigner distribution function (WDF), the propagation formula of the beam matrix in terms of second-order moments of truncated beams through atmospheric turbulence is derived. It is shown that the divergence problem of second-order moments of truncated beams can be avoided when the complex Gaussian function is applied to the WDF in the z=0 plane. The analytical expressions for second-order moments in the z=0 plane can be obtained and the accuracy can be guaranteed. And then, the beam matrix in terms of second-order moments of truncated beams in turbulence can be obtained. The results are general, i.e. the beam matrix in terms of second-order moments of non-truncated beams in turbulence and that of truncated beams in free space can be treated as two special cases of our results.