Bending (source code)
= Bending
{wiki}
\b[Bending] changes the <curvature> of an <elastic beam>, rod or sheet. In the elementary beam model, material on opposite sides of the <neutral axis> stretches and shortens. Their longitudinal <normal stresses> can balance to give zero resultant <force> while still transmitting a nonzero <bending moment>. A bent ruler illustrates why resistance to changing shape cannot be described by axial stretching alone.
For a homogeneous slender <elastic beam> in small-deflection <pure bending>, take $x$ along the beam and $y$ perpendicular to its centroidal <neutral axis>. Let $\kappa$ be the signed centreline <curvature>. The plane-section kinematics and <Hooke's law> give
$$
\varepsilon_{xx}=-\kappa y,\qquad
\sigma_{xx}=-E\kappa y.
$$
Here $E$ is <Young's modulus>, taken constant over the section. Choosing the positive <bending moment> as $M=-\int_A y\sigma_{xx}\,dA$ yields
$$
M=E\kappa\int_Ay^2\,dA=EI\kappa,
\qquad
\boxed{\sigma_{xx}=-\frac{My}{I}.}
$$
Thus the <second moment of area> $I$, rather than just the <cross-sectional area>, controls <bending stress>. The <beam flexural rigidity> is $EI$. Zero axial resultant requires $\int_Ay\,dA=0$, locating the <neutral axis> at the section <centroid> under these assumptions.<Stresses in Beams by David Roylance>[Normal Stresses, printed pp. 1–4, equations 1–7]{ref}
A <four-point bending test> makes the constant-moment region experimentally accessible. Two outer supports carry a specimen while two inner contacts apply the load. The Illinois teaching laboratory uses adjustable inner and outer spans on an Instron 4400 frame and records load and deflection electronically; it also tests notched specimens to examine <stress concentration>.<Bending Test at the University of Illinois Mechanical Testing Instructional Laboratory>[Bending Test, apparatus and notched specimens]{ref}
For an original symmetric loading example, place the outer supports at $x=0,L$ and apply total downward <force> $P$ equally at $x=a,L-a$, with $0<a<L/2$. Each support reaction is $P/2$. Between the inner contacts the <transverse shear force> is zero and the <bending moment> is constant, $M=Pa/2$. This central interval realizes the ideal beam conditions for <pure bending> away from contact disturbances. A <three-point bending test> instead has its maximum moment at the single central loading contact.
Choose $L=160\,\mathrm{mm}$, $a=40\,\mathrm{mm}$, $P=40\,\mathrm N$, rectangular width $b=10\,\mathrm{mm}$ and thickness $h=4\,\mathrm{mm}$. Then $I=bh^3/12=53.333\,\mathrm{mm^4}$ and $M=0.800\,\mathrm{N\,m}$. With stipulated $E=70\,\mathrm{GPa}$, the model gives
$$
\kappa=\frac{M}{EI}=0.2143\,\mathrm{m^{-1}},\qquad
|\varepsilon_{\mathrm{surface}}|=\frac{\kappa h}{2}=4.286\times10^{-4},\qquad
|\sigma_{\mathrm{surface}}|=30.0\,\mathrm{MPa}.
$$
These independently chosen numbers describe an assumed elastic specimen, rather than measurements of a particular material.
\Image[/four-point-bending-moment-and-strain.png]
{title=Four-point loading produces a constant <bending moment> and a linear bending <strain> profile}
{description=Original idealized fixture and exact beam-model calculation for total load 40 N, outer span 160 mm and inner contacts 40 mm from each support. The stress/strain profile is taken in the central interval, away from the contacts; the drawn beam is straight to show load positions, not its deflected shape.}
The same relation can infer a <flexural modulus> from a measured <bending moment> and <curvature>, provided the geometry and beam assumptions are adequate. Added axial <force> contributes a uniform longitudinal <normal stress> and can shift the zero-stress line. Short beams require attention to transverse <shear strain>; large rotations require a geometrically nonlinear description. The small-deflection <Euler-Bernoulli beam equation> and the more general <Euler elastica> cover different levels of that modeling.
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