= Bending stress
{title2=$\sigma_{xx}=-My/I$}
\b[Bending stress] is the longitudinal <normal stress> associated with a <bending moment> in a beam. In a homogeneous slender <elastic beam> with small deformation, plane sections remaining plane give longitudinal <strain> $\varepsilon_{xx}=-\kappa y$ relative to the <neutral axis>. Applying <Hooke's law> gives $\sigma_{xx}=-E\kappa y$.<Stresses in Beams by David Roylance>[Normal Stresses, strain and linear-elastic moment relation]{ref}
Choose the moment convention $M=-\int_A y\sigma_{xx}\,dA$. Then
$$
M=E\kappa I,\qquad I=\int_Ay^2\,dA,\qquad
\sigma_{xx}=-\frac{My}I.
$$
Here $I$ is the <second moment of area>, and the <neutral axis> passes through the section centroid when axial resultant is zero and $E$ is uniform. The sign tells which side is in tension; the magnitude grows linearly with distance from that axis.
For an original rectangular section with width $b=12\,\mathrm{mm}$ and thickness $h=6\,\mathrm{mm}$, $I=bh^3/12=216\,\mathrm{mm^4}$. A moment $M=1.2\,\mathrm{N\,m}$ produces maximum surface magnitude
$$
|\sigma_{\mathrm{surface}}|=\frac{M(h/2)}I
=16.7\,\mathrm{MPa}.
$$
The moment is $1200\,\mathrm{N\,mm}$ when using millimetre section dimensions. At fixed $M$ and $b$, doubling $h$ reduces this nominal surface stress fourfold, not merely twofold, because $I$ grows as $h^3$.
These are beam-model stresses away from local load contacts and discontinuities. Added axial force contributes $N/A$ to the longitudinal stress; a notch can produce a <stress concentration> beyond the nominal distribution. For inhomogeneous sections the neutral-axis position must be obtained from the actual stiffness distribution rather than assumed from geometry alone.
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