Natural Structural Shapes of Membrane Shells

The concept of natural structural shapes is based on the simultaneous “minimization” of the mass and the strain energy of the loaded structure, a multicriteria optimization problem. Natural structural shapes are represented by “proper” Pareto optimal designs in the sense that the limiting minimum weight and minimum stored energy designs are omitted. The method here is used to obtain optimal shell designs within the membrane theory of shells. In order to make full use of standard control theoretic methods, the problems are restricted to axisymmetrically loaded shells of revolution, a formulation involving only one independent variable. The meridional radius of curvature is used as the design variable with the Cartesian coordinates of the midsurface and a modified meridional force per unit length of the midsurface as state variables. Necessary conditions for an Edgeworth-Pareto optimum are derived and only extremal solutions based on this condition are considered.