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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.

A four-point flexural test bends a specimen using two outer supports and two inner loading contacts. For symmetric loading, the interval between the inner contacts has constant bending moment and zero transverse shear force, approximating pure bending away from contact disturbances. The Illinois laboratory uses adjustable spans and records force and deflection with a mechanical testing frame.^[ref]

Let the support span be $L$, the distance from each support to its nearest loading contact be $a$, and the total downward applied force be $P$. Each loading contact and each support carries force magnitude $P/2$. Static equilibrium givesThe central constant-moment region has length $L-2a$. Specifying whether a force symbol means total force or force at each nose prevents a factor-of-two error.

For an original example, $L=180\mathrm{mm}$, $a=60\mathrm{mm}$ and $P=80\mathrm{N}$ give reactions $40\mathrm{N}$ and central moment $2.4\mathrm{N}\mathrm{m}$. A rectangular specimen with width $b=20\mathrm{mm}$ and thickness $h=5\mathrm{mm}$ has $I=b{h}^3/12=208.33\mathrm{m}{\mathrm{m}}^4$. The nominal maximum bending stress magnitude is $M(h/2)/I=28.8\mathrm{MPa}$.

The apparatus measures force and a chosen displacement; the beam model converts them into stress, strain or flexural modulus. Midpoint displacement and displacement at the inner loading contacts are generally different. Fixture motion can additionally include machine compliance and contact indentation. Record the displacement location and both spans along with specimen dimensions.^[ref]

Unlike a central three-point load, this geometry provides a finite interval of maximum nominal moment. It tests a larger highly stressed region, so flaw distribution and where failure begins can matter when comparing results between fixtures.