On optimum thin-walled closed cross section

This paper deals with the problem of finding a minimum area thin-walled closed cross section with prescribed constant thickness and flexural rigidity. The cross section is supposed to be double symmetrical with respect to the Cartesian reference system (x0y), where 0 is the centroid of the cross section, and subjected to a bending moment M. The vector M is taken to be non-coincident with the x or y axis. This means that to represent the flexural rigidity both Ix and Iy moments of inertia are required. The function describing the centerline of the thin-walled closed cross section is taken as unknown.The solution of this problem shows that such a centerline is an ellipse.