Introduction.Finite deformations of materials composed of long, strong fibers bonded together with a weaker matrix material can be studied by using the theory of ideal fiber-reinforced materials. This theory, which is the subject of a recent book by Spencer [1], is based on the idealizations that the fibers are continuously distributed and inextensible and that the composite is incompressible in bulk. In plane deformations of such materials [2], the kinematic constraint conditions are sufficiently restrictive, so that stress-strain relations often play only a minor role. For this reason, it is relatively easy to solve problems that would be intractible if the material were isotropic [3-10]. The relation of solutions from the idealized theory to solutions for slightly extensible materials is understood fairly well [11,12], Most of the existing work on plane deformations is discussed in a recent review article [13].In the present paper we discuss deformations of cylindrical bodies with fibers lying in cross-sectional planes. The fibers may be curved, but they have the same distribution on each cross-section. The body is stretched or compressed in the axial direction and then subjected to a further plane deformation. The axial stretching causes the cross-section to change shape. The theory of superposed plane deformations is not essentially different from the theory that applies when there is no stretching [2], if the stretched state is used as the reference state. Consequently, our main object is to determine the state of deforma-