= Beam cross-section rotation
{title2=$\theta\simeq w_x$}
\b[Beam cross-section rotation] is the signed change in orientation of a material section in the bending plane. The <Euler-Bernoulli beam equation> assumes that sections remain plane and perpendicular to the deformed centreline. With a consistent angle convention, the section rotation therefore agrees with the centreline tangent angle; for a graph $w(x)$,
$$
\theta=\arctan w_x\simeq w_x.
$$
The last step is a <small-angle approximation>, with the angle measured in radians. This kinematic relation underlies the axial bending <strain>, rather than being an extra condition imposed by every support.<Stresses in Beams by David Roylance>[Normal Stresses, pp.1–2, plane-section rotation and equation1]{ref}
For an original geometric profile, take $w(x)=ax^2$ with $a=0.10\,\mathrm{m^{-1}}$. At $x=0.50\,\mathrm m$, the <slope> is $0.10$, while the exact tangent angle is $\arctan(0.10)\approx0.09967$ radians. Replacing it by $0.10$ overestimates the angle by about 0.33 percent. This is a chosen shape illustrating the approximation, not a measured equilibrium profile.
The <beam shear angle> records a possible difference between centreline slope and section rotation in a model that permits transverse <shear strain>. The Euler-Bernoulli kinematics set that difference to zero; a short or shear-compliant specimen can require a different model. A <simply supported beam> leaves its end section rotation free, whereas a <clamped boundary condition> prescribes it. Neither zero transverse displacement alone nor zero endpoint <bending moment> requires zero rotation.
Back to article page
