Governing Equations for Shells Optimization

The optimization problem consists in searching the shape and the thickness distribution of the thin shells, made of isotropic material. For the beginning, the governing equations of the thin shells are briefly summarized. The first optimization problem studies the shell of revolution for the case of pure twist moment. For this purpose, the section of shell with the normal to the symmetrical axis end planes is considered. The axial or torque loads are applied to the ends of the shell and cause the displacement of the edges. The problem consists in the optimizing the stiffness of the shell, assuming its mass to be constraint. The unknown variables are the radius of the shell and the thickness. The closed form solution of the optimization problem is obtained and corresponding isoperimetric inequality is proved.