Thick-walled hollow cylinder is an important class of engineering structure, the stress state of which depends on the loads and properties of the body materials. Under the assumptions of σθ-σr=c (σθ and σr denote the hoop stress and radial stress, respectively, c is a constant), inverse analysis of thick-walled hollow cylinder composed of functionally graded materials with uniform pressure acting on the outer surface is carried out. Analytical solutions for the Young’s modulus variation in the radial direction are obtained. It is found that only when the Young’s modulus E(r) is a specific monotone increasing function of the radius r, the pre-specified stress distribution can be satisfied. Comparing with classical homogeneous materials, stress concentration at the inner surface of hollow cylinder composed of functionally graded materials can be alleviated. Hence the elastic ultimate bearing capacity of hollow cylinder can be improved strikingly. For functionally graded materials, the elastic ultimate bearing capacity can be improved strikingly by increasing the thickness of cylinder, which is not so obvious for classical homogeneous materials.