= Moment-free natural boundary condition for a beam
{title2=$EIw_{xx}=0$}
The \b[moment-free natural boundary condition for a beam] is the zero-end-moment condition obtained when an endpoint can rotate freely and no external couple is applied there. For positive <beam flexural rigidity> $B$, it is $Bw_{xx}=0$. This differs from prescribing the endpoint <beam cross-section rotation>.
The <simply supported beam> gives the full <principle of virtual work> derivation. At its right support, the admissible <virtual displacement> $\eta$ obeys $\eta(L)=0$, but $\eta'(L)$ is arbitrary. The remaining bending boundary contribution is $B(L)w''(L)\eta'(L)$. Its vanishing for every permitted end rotation requires $B(L)w''(L)=0$. Beam virtual work pairs end moments with rotation variations and forces with displacement variations.<The Concept of Stress, Generalized Stresses and Equilibrium by Tomasz Wierzbicki>[§§3.4–3.5, equations3.46 and3.52–3.57, printedpp.3-18–3-20]{ref}
An original modified boundary example shows why the missing external couple matters. Define a signed right-end couple $m_L$ by its external <virtual work> $m_L\eta'(L)$. The boundary coefficient then becomes $Bw''-m_L$, so the free-rotation condition is $B(L)w''(L)=m_L$. With $B=2\,\mathrm{N\,m^2}$ and $m_L=0.010\,\mathrm{N\,m}$, the endpoint second derivative is $0.005\,\mathrm{m^{-1}}$, not zero. This sign definition fixes the couple convention explicitly.
Clamping instead sets $\eta'(L)=0$; that restriction permits an unknown reaction moment rather than forcing it to vanish. Finite rotational restraint leads to a <rotational spring boundary condition for a beam>. These alternatives distinguish a prescribed motion from its conjugate natural load condition.
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