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Beam root rotational compliance is the incremental rotation of a beam's attachment relative to its supporting base per increment of applied root torque. For a linear rotational restraint with stiffness $K_{\theta }>0$,The angle is measured in radians, so the compliance can be reported in $rad/(\mathrm{N}\mathrm{m})$. The root's transverse position is held fixed in this model. The rotational spring boundary condition for a beam implements this finite restraint; a clamped boundary condition is the zero-compliance limit. End moments and section rotations are conjugate quantities in beam virtual work.^[ref]

For an original static derivation, apply a transverse tip force $P$ to a beam of length $L$ and constant beam flexural rigidity $B$. The root moment has magnitude $PL$, giving ${\theta }_0=C_{\theta }PL$. In the small-angle approximation, root rotation contributes $L{\theta }_0$ to tip displacement. Bending relative to that rotated attachment gives the usual cantilever contribution, soThis derivation assumes a linear elastic beam, no root translation and an effectively rigid supporting base. The second term is fixture motion rather than a change in material Young's modulus.

For stipulated $L=0.30\mathrm{m}$, $B=6\mathrm{N}{\mathrm{m}}^2$, $C_{\theta }=0.020rad/(\mathrm{N}\mathrm{m})$ and $P=1\mathrm{N}$, the root rotates through $0.006$ radian. Root motion contributes $1.8\mathrm{mm}$, while bending contributes $1.5\mathrm{mm}$. Attributing the entire $3.3\mathrm{mm}$ to a perfectly clamped beam would infer $B=2.73\mathrm{N}{\mathrm{m}}^2$ instead of the stipulated value six.

A proposed measurement uses a known force and lever arm to apply a root moment, with optical markers on the attachment and on the supporting base to measure their relative angle. A separate displacement channel can detect unwanted root translation. Fit the small-load slope of angle against moment; using only the beam-tip displacement would mix attachment rotation with beam bending. Load reversal tests whether one linear compliance adequately describes the fixture or whether backlash and hysteresis require another model.

The 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 $B{w}_{xx}=0$. 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 ${\eta }^{\prime }(L)$ is arbitrary. The remaining bending boundary contribution is $B(L)w^{\prime \prime }(L){\eta }^{\prime }(L)$. Its vanishing for every permitted end rotation requires $B(L)w^{\prime \prime }(L)=0$. Beam virtual work pairs end moments with rotation variations and forces with displacement variations.^[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 $m_L{\eta }^{\prime }(L)$. The boundary coefficient then becomes $B{w}^{\prime \prime }-m_L$, so the free-rotation condition is $B(L)w^{\prime \prime }(L)=m_L$. With $B=2\mathrm{N}{\mathrm{m}}^2$ and $m_L=0.010\mathrm{N}\mathrm{m}$, the endpoint second derivative is $0.005{\mathrm{m}}^{-1}$, not zero. This sign definition fixes the couple convention explicitly.

Clamping instead sets ${\eta }^{\prime }(L)=0$; 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.