 Circular bending energy of an elastic filament

A circular loop of radius $R$ has constant curvature $1/R$ and contour length $2\pi R$. Integrating the quadratic bending energy density $K_b/(2R^2)$ gives the displayed energy. With $K_b=k_{B}TA$ this is $\pi k_{B}TA/R$. Applied-force work, axial strain, twist and contact energies are separate terms; the loop formula alone does not determine an actual loaded loop's radius.