 Solution

With endpoints held at zero displacement but free to rotate, variation permits arbitrary endpoint $\delta {\psi }_x$ and gives the natural zero-moment condition ${\psi }_{xx}=0$ there. These are simply supported endpoints. The normal modes areThe first mode becomes unstable when $F+B(\pi /l{)}^2<0$, soEquivalently the compressive load exceeds $|F{|}_{\mathrm{crit}}={\pi }^{2}B/l^2$. At equality the linear mode is neutral and nonlinear terms are needed to determine its post-buckling amplitude. The quadratic energy of $A\sin (\pi x/l)$ is $A^{2}l[F(\pi /l{)}^2+B(\pi /l{)}^4]/4$, giving the same sign test.

The endpoint displacement condition alone does not impose clamping. The quoted threshold uses the natural free-slope boundary condition of the stated variational problem. If rotations were also fixed, the admissible modes and threshold would change; one must not import a clamped-end formula into this result.