Ideal stretch forming for minimum weight axisymmetric shell structures

A companion paper “Minimum Weight Design of Axisymmetric Shell Structures” by Richmond and Azarkhin describes the design of axisymmetric thin-walled structures and parts that can support specified loads with minimum material. The result is an optimum shape and thickness distribution in the final part when the strength of the material is assumed to be uniform. Here, we demonstrate the application of ideal forming theory to design sheet stretching processes that can produce the optimum shapes and thickness distributions from flat sheets of uniform thickness. Specific designs are achieved for producing minimum weight shell structures that will support a specified uniform pressure assuming both the Mises and the Tresca yield criteria along with the rigid-perfectly plastic flow condition. In the case of the Tresca yield condition, the optimum structure is a spherical shell segment with uniform thickness, and an associated ideal stretching process is hydraulic bulging. Because the effects of strain hardening have been neglected in the structural optimization theory, it has been possible here to design the minimum weight structure and its forming process sequentially. In subsequent work, we plan to include the effects of strain hardening on the shell strength, which will then require coupled design of the structure and its forming process. Also extension of these methods to three-dimensional geometries will be considered.