In stiffness limited applications, it is elastic deflection that is the active constraint: it limits performance. In strength limited applications, deflection is acceptable provided the component does not fail; strength is the active constraint. Derive the material index for selecting materials for a beam of length \(L\), specified strength and minimum weight. For simplicity, assume the beam to have a solid square cross-section \(t \times t\). You will need the equation for the failure load of a beam (Appendix B, Section B4). It is

\[F_{f}=\frac{I \sigma_{f}}{\gamma_{m} L}\]
where \(\gamma_{m}\) is the distance between the neutral axis of the beam and its outer filament and \(I=t^{4} / 12=A^{2} / 12\) is the second moment of the cross-section. The table given in this exercise itemizes the design requirements.
Plot the criterion on a copy of the \(\left(\sigma_{f}-ho\right.\) ) chart of Figure 3.4 and use it to identify promising candidate materials. (If you have access to the CES Edu software, make the chart and apply the index to find candidate materials.)

Data From FIg. 3.4

Data From Appendix B4

Function Constraints Objective Free variables Beam Length L is specified Beam must support a bending load F without yield or fracture Minimize the mass of the beam Cross-section area, A Choice of material