= Beam flexural rigidity
{title2=$B=EI$}
\b[Beam flexural rigidity] is the coefficient relating a slender beam's <bending moment> to its <curvature> in linear bending. For a homogeneous material with uniform <Young's modulus> $E$ over the material area,
$$
B=EI,\qquad I=\int_A z^2\,dA,
$$
where $z$ is distance from the appropriate <neutral axis>. The magnitude relation is $|M|=B|\kappa|$; signs depend on the chosen moment and curvature conventions. Integrating the linear bending <stress> over the section gives this coefficient.<Stresses in Beams by David Roylance>[Normal Stresses, pp.2–4, equations3–6]{ref}
For an original rectangular example, choose width $b=10\,\mathrm{mm}$, bending thickness $h=2\,\mathrm{mm}$ and $E=70\,\mathrm{GPa}$. The <second moment of area> is $I=bh^3/12=6.667\times10^{-12}\,\mathrm{m^4}$, giving $B\approx0.4667\,\mathrm{N\,m^2}$. Doubling only the thickness multiplies $B$ by eight. Rotating the original rectangle by ninety degrees about its length instead exchanges width and thickness: $I$ increases by $(b/h)^2=25$ even though material area and modulus are unchanged.
Thus a bending experiment measures a combination of material response and geometry. Extracting $E$ requires the correct bending axis and <second moment of area>; the <three-point flexural test> is one route. This beam coefficient has units $\mathrm{N\,m^2}$, distinct from the plate coefficient called <bending stiffness>. If modulus varies across a section, the relevant stiffness requires a material-weighted area integral and the appropriate <neutral axis>, rather than an arbitrary average modulus multiplied by $I$.
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