Example 2: Selection of Material for a Light and Stiff Beam.

Function:                    Beam to withstand a bending load of F
Objective:                   Minimise mass (m) where , where is the material density.
Constraints:               (i) Length L is specified, (ii) Must not bend under bending load, F
Free variable:            (i) Edge length, b, (ii) Material
Performance Equation:       
The Performance Equation can be developed considering the fact that the beam must be stiff enough to allow a maximum critical deflection, , under the bending load, F. Thus, the Performance Equation can be given as

(2)

where is the maximum permissible deflection, E is the young’s modulus, I is the second moment of area. The stiffness, S, of the beam, can be written as,  and the second moment of area, I, can be written as, .

The Performance Equation can now be rewritten by substituting one of the free variables (edge length, b) as

(3)

The material index, M₂, in this case becomes  and a material with higher value of M₂ is expected to perform better in comparison to a material with lower value of M₂. In other words, the Material Index (M₂) will depict if a material with higher weight (density) has to be selected to ensure that the same has sufficient stiffness (i.e. E) to avoid bending during service.