Problem $3\mathrm{.}17$

Problem Statement: A column is a structural member that resists a compressive force. If a column

fails, it does so by buckling $-$ that is$,$ by suddenly deflecting sideways. For relatively long, thin

columns, this occurs at a critical load of

$P_{cr}=\frac{{}^{2}EI}{L^2}$

where $E$ is the elastic modulus of the column material and I is the moment of inertia of the cross$-$

sectional area. Assume that the cross section is a thin$-$walled tube of wall thickness $t$ and inner

radius $r_1,$ with the proportions $t=0\mathrm{.}2r_1$ being maintained, as the section size is allowed to vary

with material choice. The column must have a particular length $L$ and resist a load $P$ that is lower

than $P_{cr}$ by a safety factor $x.$ Noting that the relative importance of light weight and cost may vary

with the application, make a preliminary selection of column materials from Table $3\mathrm{.}13$ for the

following:

a$)$ A structural compression member in a space station.

b$)$ A support for the second floor above the garage in a private residence.