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A flexural modulus is a stiffness inferred from a bending experiment using a specified stress–strain or force–deflection model. It has units of pressure. In a homogeneous, isotropic, linearly elastic slender beam with negligible shear and fixture deformation, it agrees with the Young's modulus entering $M=EI\kappa$. It is not the flexural strength or the load at fracture.

For a simply supported uniform beam in a symmetric four-point flexural test, let outer span be $L$, loading contacts lie at $a$ and $L-a$, and $P$ be their total force. Integration of the small-deflection Euler-Bernoulli beam equation gives midpoint deflection magnitudeEquivalently, with measured linear slope $S=dP/d\delta$,The formula uses midpoint displacement relative to the supports, not automatically the testing machine's crosshead displacement. The underlying moment-curvature law is the beam relation derived by Roylance; actual test conversion additionally requires the fixture spans and specimen geometry.^[ref][ref]

For an original rectangular example, choose $L=120\mathrm{mm}$, $a=30\mathrm{mm}$, $b=10\mathrm{mm}$ and $h=4\mathrm{mm}$, so $I=53.333\mathrm{m}{\mathrm{m}}^4$. A corrected midpoint slope $S=100\mathrm{N}\mathrm{m}{\mathrm{m}}^{-1}$ gives $E_{\mathrm{flex}}\approx 46.4\mathrm{GPa}$. This is an inferred model value from stipulated inputs, not a tabulated property of a named material.

If measured displacement includes an extra compliance $C_{m}P$, the measured compliance is $d{\delta }_{\mathrm{meas}}/dP=C_{\mathrm{beam}}+C_m$. Ignoring positive fixture compliance reduces the apparent inferred modulus. Transverse shear, anisotropy, nonlinear response and time-dependent material behavior can also make flexural and tensile measurements differ. A reported value should therefore identify geometry, slope convention, displacement measurement and loading conditions.

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 giveHere $y$ 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.^[ref]

For an original example, let $M=2\mathrm{N}\mathrm{m}$ and $EI=50\mathrm{N}{\mathrm{m}}^2$. 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 $4\times 10^{-5}$. 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.