Ex Example. Optimal design of a beam.

Example. Optimal design of a composite. The best structure to resist a one dimensional loading is laminate. It remain the optimal structure for any close to one-dimensional loading.

AN Example: Optimal design of a structure is placed below, in the Epiloque

Suppose that the the optimal composite is the mixture of a material and the void. /TR>.
The best structure against a uniaxial loading is again the laminate: Picture The project is extremely unstable to any other loading: it is collapsing as the deck of cards if any loading different from the uniaxial is applied.

Suppose that admissible loadings are the following: Picture Picture Picture
The structure stable to the following six loadings is found from the conditions that the stiffness against five of the loadings (## 1, 3, 4, 5, 6) is equal. These conditions require the symmetry of structure and they lead to the following design. Note that the extremum is non-smooth function of the structural parameters.

Similarly, optimal design of a wheel requires the axial symmetry of the design in spite of the non-symmetry of the loading.