Performance Index & Material Index

• The design of a structural element involves minimization or maximization of an index given by p=f[F,G,M]
• F - Functional Requirement G - Geometric Requirement M - Material Requirement
• For certain designs - variable separation exist :
  I E p=f ₁ [F] f ₂ [G] f ₃ [M]
• Here M denotes the Material Index

Example: Material Index(M) for a light stiff Tie

• Performance Index - Minimize the mass of the Tie - m=Al ρ where A is the area of cross-section of the Tie, l - the length and ρ the density.
• Constraint - The deflection of the tie should be within the permissible limit - Fl/AE< δ
• Eliminating A , we get m>(F / δ )(l ² )( ρ /E)
• Lightest Element that can carry F is the one which has maximum E/ ρ . Material Index M= E/ ρ

Similar Material Indices

• Light Stiff Beam:
  Free area ( no constraint on height and width) - M = E ^1/2 / ρ
  Free Height - M = E ^1/3 / ρ
  Free width - M = E/ ρ
• Light Strong Beam: M= σ_f ^3/2 / ρ
• Cheap Stiff Coloumn
  C=C _m Al ρ , F<n π ² EI/l ²
  M = E ^1/2 / C _m ρ