The best choice of material for a light strong column depends on its aspect ratio: the ratio of its height \(H\) to its diameter \(D\). This is because short, fat columns fail by crushing; tall slender columns buckle instead. Derive two performance equations for the material cost of a column of solid circular section and specified height \(H\), designed to support a load \(F\) large compared to its selfload, the first using the constraints that the column must not crush, the second that it must not buckle. The table summarizes the needs.

a. Proceed as follows
1. Write an expression for the material cost of the column - its mass times its cost per unit mass, \(C_{m}\).
2. Express the two constraints as equations and use them to substitute for the free variable, \(D\), to find the cost of the column that will just support the load without failing by either mechanism 3. Identify the material indices \(M_{1}\) and \(M_{2}\) that enter the two equations for the mass, showing that they are \[M_{1}=\left(\frac{C_{m} ho}{\sigma_{c}}\right) \quad \text { and } \quad M_{2}=\left[\frac{C_{m} ho}{E^{1 / 2}}\right]\]
where \(C_{m}\) is the material cost per \(\mathrm{kg}, ho\) the material density, \(\sigma_{c}\) its crushing strength and \(E\) its modulus.

b. Data for six possible candidates for the column are listed in the table with this problem. Use these to identify candidate materials when \(F=10^{5} \mathrm{~N}\) and \(H=3 \mathrm{~m}\). Ceramics are admissible here, because they have high strength in compression.
Data for candidate materials for the column

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Function Constraints Objective Free variables Column Must not fail by compressive crushing Must not buckle Height H and compressive load F specified. Minimize material cost C Diameter D Choice of material