Axially symmetric membranes composed of inextensible fibers are considered. When the shape of the membrane and the tensile strength of the fibers are given, the minimum weight of fiber which can support a given internal pressure is proportional to the volume enclosed by the membrane. Minimum weight designs are isotensoid, i.e., every fiber is under the same tension. Every isotensoid design is a minimum weight design. Particular attention is devoted to geodesic isotensoid designs, in which fibers are required to lie along geodesics on the membrane. The equation relating the shape of the membrane to the distribution of fibers on it is obtained. When the shape is given, this equation is an integral equation for the fiber distribution, which is solved by using the Laplace transform. Illustrative examples involving cones, spheres, ellipsoids, and cylinders are solved.

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