Materials such as resins reinforced by strong aligned fibres exhibit highly anisotropic elastic behaviour in a sense that their elastic moduli for extension in the fibre direction are frequently of the order of 50 or more times greater than their elastic moduli in transverse extension or in shear. The mechanical behaviour of many fibre-reinforced composite materials is adequately modelled by the theory of linear elasticity for transversely isotropic materials, with the preferred direction coinciding with the fibre direction. In such composites the fibres are usually arranged in parallel straight lines. However, other configurations are used. An example is that of circumferential reinforcement, for which the fibres are arranged in concentric circles, giving strength and stiffness in the tangential (or hoop) direction. The theory of strongly anisotropic materials has been extensively discussed in the literature; Belfield et al.[1] studied the stress in elastic plates reinforced by fibres lying in concentric circles.