= Rotational spring boundary condition for a beam
{title2=$Bw_{xx}(L)+K_\theta\theta(L)=m_L$}
A \b[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 is
$$
B(L)w''(L)+K_\theta\theta(L)-m_L=0.
$$
Here $B=EI$ 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>.<The Concept of Stress, Generalized Stresses and Equilibrium by Tomasz Wierzbicki>[§§3.4–3.5, equations3.46 and3.52–3.57]{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, $-B(0)w''(0)+K_\theta\theta(0)=0$.
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 $PL^3/(3B)$ to the tip displacement; rigid root rotation contributes $PL^2/K_\theta$. Thus
$$
w(L)=\frac{PL^3}{3B}+\frac{PL^2}{K_\theta}.
$$
With $P=1\,\mathrm N$, $L=0.20\,\mathrm m$, $B=2\,\mathrm{N\,m^2}$ 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.
Back to article page
