Abstract While many shell theories and solutions are available to the designer for conventional pressure-vessel analysis, relatively little work exists for composite material vessels. Such vessels are in general constructed of thin layers consisting of filaments (reinforcing fibers) and matrix materials which may have different thicknesses and different fiber arrangements. Because of this construction, composite vessels must be analyzed according to theories which allow for nonhomogeneous, anisotropic material behavior. These theories are, however, difficult to derive by means of the classical method. A way of departing from the classical assumptions is to apply the method of asymptotic integration of the three-dimensional elasticity equations [1–3]. This method has, as its foundation, the desire to obtain rational two-dimensional theories which are approximately valid when a problem parameter is very small.