Metal Foams: A Design Guide

58 Metal Foams: A Design Guide
(i) Write down an equation for the objective
(ii) Eliminate the free variable(s) in this equation by using the constraints
(iii) Read off the grouping of material properties (called the material index)
which maximize or minimize the objective.
A more detailed recipe is given in Table 5.2. Indices for numerous standard
specifications are listed in the Appendix at the end of this Design Guide.
5.3 Two examples of single-objective optimization
Panel of specified stiffness and minimum mass
The mode of loading which most commonly dominates in engineering is not
tension, but bending. Consider the performance metric for a panel of specified
length, ℓ, and width, b (Figure 5.1), and specified stiffness, with the objective
of minimizing its mass, m. Themassis
t
m D bt ⊲5.1⊳
l
F/unit width
Figure 5.1 A panel of length, ℓ, width, b, and thickness, t, loaded in
bending by a force, F, per unit width
where t is the thickness of the panel and is the density of the material of
which it is made. The length, ℓ, width, b, and force, F, per unit width are
specified; the thickness, t, is free. We can reduce the mass by reducing t, but
there is a lower limit set by the requirement that the panel must meet the
constraint on its bending stiffness, S, meaning that it must not deflect more
than υ under a load Fb. To achieve this we require that
S D Fb
υ
B1EI
D ½ SŁ
3
ℓ
b
⊲5.2⊳
where S Ł is the desired bending stiffness, E is Young’s modulus, B1 is a
constant which depends on the distribution of load (tabulated in Chapter 6,
Section 6.3) and I is the second moment of the area of the section. This, for
a panel of section b ð t, is
I D bt3
12
⊲5.3⊳