A modified Galerkin approach is employed to calculate the dynamic characteristics of a complex non-uniform fluid-conveying pipe assembled by a uniform and a conical segment. The effects of two geometric parameters (length ratio of the uniform part and conical truncation factor) on the dynamic stability are studied. Results prove that for the assembled fluid-conveying pipe 1) when fluid enters the pipe from the thinner end, the natural frequencies are higher than those when it enters from the wider end; 2) the critical flow velocity increases linearly with the increase of the uniform part ratio, while it decreases squarely with the increase of the conical truncation factor in a limited range and 3) the clamp-pined non-uniform pipe has a higher dimensionless critical velocity than the pin-clamped pipe when fluid enters from the wider end.