The paper is devoted to the analytic optimization of the thickness distribution of axisymmetric vessels where the cost functional to minimize is the compliance. The goal is achieved by minimizing the strain energy under mass constraint. The theoretical framework of the mechanics of thin walled membrane shells is recalled at the beginning of the paper. The optimization problem is stated, formulated and solved by means of calculus of variations. Based on the augmented Lagrangian formalism, the Euler-Lagrange equation leads to analytical optimal thickness distributions. Numerical illustrative examples are given and the optimal solutions are obtained for standard and nonstandard meridian shapes and end closures. Finally, the compliance of optimal solutions is compared with its counterpart of constant-thickness vessels of the same mass.
