Euler buckling of an elastic filament

A pinned filament of length $L$ and bending modulus $A$ first buckles under compressive force $F_c={\pi }^{2}A/L^2$.

Thermal fluctuations smooth the zero-temperature pitchfork in filament buckling: the mean compression remains analytic and nonzero below the mechanical threshold, with rounding controlled by $L/L_p$.

A vertical rod can buckle under its own distributed weight. For a rod clamped below and free above, the threshold is set by a Bessel-function eigenvalue rather than the sinusoidal mode of a rod under constant end load.