Beam Strain⚠ unverified

Understanding the Parameters

• Distance from the neutral axis $y$ — the through-depth position of the fibre. At $y=0$ (the neutral axis) the fibre neither stretches nor compresses, so strain is zero; strain grows linearly to its extremes at the top and bottom surfaces. The sign of $y$ distinguishes the tension side from the compression side.

• Radius of curvature $\rho$ — how sharply the beam is bent. A large $\rho$ means a gentle curve and small strains; as $\rho \to \infty$ the beam is straight and $\varepsilon \to 0$. The reciprocal $\kappa = 1/\rho$ is the curvature, so the relation is equivalently $\varepsilon = \kappa y$ — strain equals curvature times distance.

• Strain $\varepsilon$ — dimensionless (metre per metre). For elastic design it stays well below ~0.002 for steel; the linear-through-depth shape is the key result, not the magnitude.

Derivation (Approaching a Proof)

Consider a short length of beam subtending a small angle $\mathrm{d}\theta$ at the centre of curvature. Under the Euler–Bernoulli assumption that plane cross-sections remain plane and perpendicular to the neutral axis, each section simply rotates about the neutral axis.

• The fibre on the neutral axis has length $\mathrm{d}s = \rho\, \mathrm{d}\theta$ and, by definition of the neutral axis, does not change length.

• A fibre at distance $y$ lies on a radius $(\rho + y)$ (measuring $y$ positive away from the centre of curvature), so its deformed length is $(\rho + y)\,\mathrm{d}\theta$.

The strain is the change in length over the original length:

$$\varepsilon = \frac{(\rho + y)\,\mathrm{d}\theta - \rho\,\mathrm{d}\theta}{\rho\,\mathrm{d}\theta} = \frac{y\,\mathrm{d}\theta}{\rho\,\mathrm{d}\theta} = \frac{y}{\rho}.$$

No material property enters — this is pure geometry (kinematics). Combining it with Hooke's law $\sigma = E\varepsilon$ and moment equilibrium yields the flexure formula (Beam Bending Stress) and the moment–curvature relation $M = EI/\rho$.

Dimensional check. $[\varepsilon] = \text{m}/\text{m} = 1$ (dimensionless). ✓

History and Development

The plane-sections kinematic hypothesis dates to Jacob Bernoulli (1690s) and was refined by Euler and Daniel Bernoulli into the elastica theory. Navier (1820s) tied the strain distribution to the elastic stress and moment, completing engineering beam theory. The linear strain profile is confirmed daily by strain gauges on real beams and is the basis for interpreting bending tests.

Related Concepts: Beam Bending Stress, Rectangular Moment of Inertia, Hooke's Law strain, Section Modulus, Continuous Beam Moment

Notes: Valid for small deflections and the plane-sections assumption. The unused epsilon input is a registry artefact (see note). Curvature $\kappa = 1/\rho$; the moment–curvature link is $\kappa = M/(EI)$.