= Pure bending
{title2=$N=V=0,\quad M=\mathrm{constant}$}
\b[Pure bending] is a beam loading state with zero axial resultant $N$, zero <transverse shear force> $V$ and a nonzero constant <bending moment> $M$ over the interval considered. Equal and opposite end couples provide an ideal realization. Zero net force on a section does not mean zero local stress: opposing longitudinal tensions and compressions can form a couple.
For a homogeneous slender <elastic beam>, small-deformation kinematics and <Hooke's law> give
$$
\varepsilon_{xx}=-\kappa y,\qquad
\sigma_{xx}=-E\kappa y,\qquad M=EI\kappa.
$$
Here $y$ is measured from the centroidal <neutral axis>, $I$ is the <second moment of area>, $E$ is <Young's modulus> and $\kappa$ is curvature. Uniform $E$, $I$ and $M$ produce uniform curvature. The geometric assumptions and moment sign convention are those of the elementary <bending stress> model.<Stresses in Beams by David Roylance>[Normal Stresses, plane-section strain and moment-curvature relation]{ref}
For an original example, let $M=2\,\mathrm{N\,m}$ and $EI=50\,\mathrm{N\,m^2}$. Then $\kappa=0.04\,\mathrm{m^{-1}}$. Across a $0.10\,\mathrm m$ interval the tangent direction changes by approximately $\kappa\ell=0.004$ radians. At $1\,\mathrm{mm}$ from the neutral axis, the strain magnitude is $4\times10^{-5}$. The small-slope sag relative to the interval's chord is $\kappa\ell^2/8=50\,\mu\mathrm m$.
The central region of a symmetric <four-point flexural test> is an experimental approximation: the external load contacts lie outside a constant-moment interval. Local contact disturbances remain near the loading noses, and transverse shear exists in the outer spans. Calling the central interval pure bending therefore specifies a region and a model; it does not claim that the entire fixture has zero force or shear.
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