Flexural modulus
= Flexural modulus
{title2=$E_{\mathrm{flex}}$}
A \b[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 magnitude
$$
\delta=\frac{Pa(3L^2-4a^2)}{48EI}.
$$
Equivalently, with measured linear slope $S=dP/d\delta$,
$$
E_{\mathrm{flex}}=\frac{a(3L^2-4a^2)}{48I}S.
$$
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.<Stresses in Beams by David Roylance>[Normal Stresses, moment-curvature law]{ref}<Flex Testing by Instron>[span and specimen dimensions in flexural calculations]{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 $I=53.333\,\mathrm{mm^4}$. A corrected midpoint slope $S=100\,\mathrm{N\,mm^{-1}}$ gives $E_{\mathrm{flex}}\approx46.4\,\mathrm{GPa}$. This is an inferred model value from stipulated inputs, not a tabulated property of a named material.
If measured displacement includes an extra compliance $C_mP$, 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.