Material index for a cheap, stiff column (Figure E6).

In the last two exercises the objective has been that of minimizing weight. There are many others. In the selection of a material for a spring, the objective is that of maximizing the elastic energy it can store. In seeking materials for thermal-efficient insulation for a furnace, the best are those with the lowest thermal conductivity and heat capacity. And most common of all is the wish to minimize cost. So here is an example involving cost.

Columns support compressive loads: the legs of a table; the pillars of the Parthenon. Derive the index for selecting materials for the cheapest cylindrical column of specified height, H, that will safely support a load F without buckling elastically. You will need the equation for the load Fcrit at which a slender column buckles. It is

2 2

Fcrit = n 2E I

H

where n is a constant that depends on the end constraints and

I =r⁴ / 4 = A² / 4 is the second moment of area of the column(see UASSPfor both). The table lists the requirements.

Function	Cylindrical column
Constraints	Length L is specified Column must support a compressive load F without buckling
Objective	Minimize the material cost of the column
Free variables	Cross-section area, A Choice of material