Flexural modulus ($E_{\mathrm{flex}}$)

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.