Challenge Level

When you apply a steadily increasing axial load to an initially-straight strut after a while it will start to bend, and then, when a critical load $P$ is reached, it will buckle. You could try it with a matchstick.

Euler showed that at the point of buckling the strut is in a static equilibrium state: like a ball balanced at the top of a slope where the slightest push will cause it to roll down. At each point along the beam the moment due to the bending stiffness and the moment due to the axial force being applied are perfectly in balance, and if you increase the load by just a tiny bit it will break.

By considering a general position $x$ on the deformed strut, you can derive Euler's basic formula for $P$ by constructing and solving a second order differential equation.

You need to know two new engineering formulae: The moment due to stiffness is $M = B \kappa$, where $B$ is bending stiffness (a property of the beam) and $\kappa$ is curvature. You also need to know that we can approximate $\kappa$ as $-\frac{d^2v}{dx^2}$, where $v$ is displacement in the direction perpendicular to the initial direction of the beam (see Beam Me Up for a derivation of this).