Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 355 / 1 / c / Solution

The transverse force density is $f=-A{h}_{xxxx}=\zeta h_t$. Its resultant isby the free-end shear conditions. To linear order, its torque about $x=0$ isbecause both end curvatures equal the same imposed ${\kappa }_0$. Thus every decaying mode is force-free and torque-free, consistently leaving the translational and rotational zero modes unchanged.