Buckling: many structural and machine parts support a uniform compressive load across the cross section. Depending on the shape, failure often occurs well before the loading exceeds the limiting stress on the material. Instead, the part buckles under the load.

For parts under a compressive load, you need to check the load limit for buckling.

Here, we will start with the most basic form $-$ the Euler formula with modified end constants. This formula gives you the critical load $P_{er}$ for a part with known dimensions and material. The strength of the material is not a factor here $-$ only the modulus of elasticity.

Before proceeding, review the document where the name starts with $3020$ Buckling. This is a $2$ page summary using material from the machine design textbook. What you'll need here is mostly on page $1.$ On page $2,$ do look at the equation for finding the size needed to support a given load $($eqn $4-46)$ and the slenderness criteria. For the moment, do not worry about the J$.$B$.$ Johnson equation.

This source refers to a pinned end connection as rounded.

Now $-$ see what you can do with a problem. Draw the entire column with the critical load.

Find the critical load $P_0$ for the column described below:

Length $=36$ inches

Circular Cross Section; Diameter $=1$ inch

Material: Steel $E=3010$

ENGT $3020$ Statics and Strength of Materials $-5^{th}$ Homework Assignment

Do this for each of the four sets of end conditions shown. Use the recommended value of $C$ in each case.

Do check to confirm that your units are consistent. Your result for the critical load $($force$)$ should come out here in pounds $(l{b}_f).$

While you are at it$,$ also find the radius of gyration $k($defined between eqns $4-43$ and $4-44)$ for this column.

Next, find the slenderness ratio $\frac{1}{k},$ where $1$ is the length.

Find the Slenderness Criteria $($eqn $4-45)$ an A$36$ structural steel with a yield stress S$,$ of $36,000$ psi. For $C,$ use the recommended value given for rounded$-$rounded end conditions. Use this to tell if the long, slender column model is appropriate.

Finally, find the compressive normal stress at the critical load. That's what you have been doing for tensile loads; F$/$A$.$ For compression the sign is negative.