This paper derives the minimum-weight shapes of head closures for cylindrical pressure vessels. Head configurations, weights, and membrane stresses are obtained for thin-shell heads of isotropic materials with uniform internal pressure by restricting the knuckle shear stresses to constant values where possible and by minimizing the weight of prescribed-shape heads. Torispherical, ellipsoidal, and Biezeno-type heads with constant and varying thickness (neglecting the effects of bending) are investigated. Enclosed-volume constancy and supporting structures are also considered. Certain highly efficient shapes are obtained that weigh up to 11 per cent less than the corresponding hemispherical capping closure.

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