This paper presents an exact solution for an inhomogeneous, transversely isotropic, elastic circular cylinder subjected to axisymmetric force and displacement boundary conditions. The solution is obtained on the basis of an eigen-function expansion method and can satisfy all the boundary conditions prescribed on the curved and end surfaces of the cylinder. It can be used directly in the micromechanical analysis of fiber reinforced composites to investigate the typical Representative Volume Element (RVE). The element consists of a combined circular cylinder composed of a solid inner circular cylinder of transversely isotropic fiber and a concentric outer circular cylinder of isotropic matrix material. Using this solution, all the stress and displacement components of both the inner fiber and the outer matrix, and hence the stress transfer in the interface between the fiber and matrix, are expressed analytically. The numerical results presented show that stress concentration occurs near the ends of the cylinder where external forces are applied.