Ex
<B>Example. Optimal design of a beam</B>. 
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<B>Example. Optimal design of  a composite.  </B>

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




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AN Example: Optimal design of  a structure is placed below, in the Epiloque
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Suppose that the  the optimal composite is the mixture of a material and the void. 

<TABLE BORDER=0 ><TR><TD>The best structure against a uniaxial loading is again the laminate:
</TD><TD>Picture</TD>
<TD>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.</TD>/TR>.
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<TABLE BORDER=0 <TR><TD>Suppose that admissible loadings are the following:
</TD> <TD>Picture</TD> <TD>Picture</TD> <TD>Picture</TD> </TR></TABLE>
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. 

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