Materials selection—the basics

Attribute limits and material indices
Constraints set property limits. Objectives define material indices. When the objective in not coupled to a constraint, the material index is a simple material property. When, instead, they are coupled, the index becomes a group of properties
Material indices when objectives are coupled to constraints
Figure 5.7. The loading on a component can generally be decomposed into some combination of axial tension, bending, torsion, and compression. Almost always, one mode dominates. So common is this that the functional name given to the component describes the way it is loaded: ties carry tensile loads; beams carry bending moments; shafts carry torques; and columns carry compressive axial loads. The words ‘‘tie’’, ‘‘beam’’, ‘‘shaft’’, and ‘‘column’’ each imply a function.

Material index for a light, strong tie-rod. A design calls for a cylindrical tie rod of specified length L to carry a tensile force F without failure; it is to be of minimum mass, as in the uppermost sketch in Figure 5.7. The length L is specified but the cross-section area A is not. Here, ‘‘maximizing performance’’ means ‘‘minimizing the mass while still carrying the load F safely’’. The design requirements, translated, are listed in Table 5.4.

We first seek an equation describing the quantity to be maximized or minimized. Here it is the mass m of the tie, and it is a minimum that we seek. This equation, called the objective function, is

where A is the area of the cross-section and is the density of the material of which it is made. The length L and force F are specified and are therefore fixed; the cross-section A, is free. We can reduce the mass by reducing the cross-section, but there is a constraint: the section-area A must be sufficient to carry the tensile load F, requiring that