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A rotational spring boundary condition for a beam lets an endpoint rotate while imposing a restoring torque proportional to that rotation relative to its support. Let $K_{\theta }>0$ be the spring stiffness, with zero rest angle. Its stored elastic energy is $K_{\theta }{\theta }^2/2$, where the small beam cross-section rotation is $\theta =w_x$.

At a right endpoint whose transverse position is fixed, add this energy and the applied-couple potential $-m_L\theta (L)$ to the bending energy. The coefficient of the free rotation variation isHere $B=EI$ is beam flexural rigidity, and $m_L$ is defined by external virtual work $m_L\delta \theta (L)$. End moment and rotation are conjugate variables in the beam principle of virtual work.^[ref] The spring term and the following example are independently derived. At the left endpoint, the bending boundary term has the opposite sign: with no applied end couple, $-B(0)w^{\prime \prime }(0)+K_{\theta }\theta (0)=0$.

For an original fixture model, hold the left endpoint position fixed, restrain its rotation by the spring, and apply a downward end force $P$ at distance $L$. The root moment magnitude is $PL$, giving root rotation $PL/K_{\theta }$. Bending relative to that rotated root contributes $P{L}^3/(3B)$ to the tip displacement; rigid root rotation contributes $P{L}^2/K_{\theta }$. ThusWith $P=1\mathrm{N}$, $L=0.20\mathrm{m}$, $B=2\mathrm{N}{\mathrm{m}}^2$ and $K_{\theta }=20\mathrm{N}m/rad$, these contributions are $1.333$ and $2.000\mathrm{mm}$, giving $3.333\mathrm{mm}$ total. Root rotation is $0.010$ radian, within the stipulated small-rotation model.

As $K_{\theta }$ increases without bound, the root approaches a clamped boundary condition. At zero stiffness this particular end-loaded arrangement becomes a rotational mechanism; it has no finite static deflection for nonzero $PL$. A small-couple fixture check can estimate $K_{\theta }$ from measured moment and angle. The resulting beam root rotational compliance is distinct from deformation of the beam material.