Wing Bending Stress⚠ unverified

Understanding the Parameters

• Bending moment $M$ — the internal moment at the section, set by the load and geometry. For a wing in steady flight it grows from near zero at the tip to a maximum at the root, roughly as the integral of the lift distribution times its moment arm — which is why wing roots are the most heavily built part of the structure. $M$ scales with load factor, so a $2.5\,g$ manoeuvre bends the wing $2.5\times$ as hard as level flight.

• Distance from the neutral axis $y$ — bending produces a linear stress distribution: zero at the neutral axis, tensile on one face, compressive on the other, peaking at the extreme fibre $y = c$. The upper skin of a wing goes into compression and the lower into tension under positive (up) bending. Because stress is proportional to $y$, material near the neutral axis carries little load — hence spar caps placed at top and bottom.

• Second moment of area $I$ — the section's geometric stiffness against bending, with units m⁴. It rewards placing area far from the neutral axis ($I = \int y^2\,dA$), so a tall, deep wing box is dramatically stiffer and stronger in bending than a thin one of the same area. The ratio $I/c = Z$ is the section modulus, the single number that captures a section's bending strength.

• The output $\sigma$ — the direct stress from bending alone; real spar caps also carry axial and thermal stresses that superpose. Compare $\sigma$ to the allowable via a margin of safety.

Derivation (Approaching a Proof)

The flexure formula follows from Euler–Bernoulli beam theory and its central kinematic assumption: plane cross-sections remain plane and perpendicular to the neutral axis as the beam bends. Under this assumption a fibre at distance $y$ from the neutral axis stretches in proportion to $y$, so the strain varies linearly,

$$\varepsilon(y) = \frac{y}{R},$$

where $R$ is the radius of curvature of the bent beam. For linear-elastic material (Hooke's law) the stress is likewise linear,

$$\sigma(y) = E\,\varepsilon = \frac{E\,y}{R}.$$

Internal stresses must reproduce the applied moment. Summing the moment of the stress distribution about the neutral axis,

$$M = \int_A \sigma\,y\,dA = \frac{E}{R}\int_A y^2\,dA = \frac{E\,I}{R},$$

which identifies the curvature $1/R = M/(EI)$. Substituting back to eliminate $R$:

$$\sigma(y) = \frac{E\,y}{R} = \frac{E\,y}{1}\cdot\frac{M}{EI} = \frac{M\,y}{I}. \qquad\blacksquare$$

The neutral axis passes through the centroid of the section (the requirement that net axial force be zero when there is pure bending forces $\int y\,dA = 0$, i.e. the centroidal axis).

Dimensional check. $$\left[\frac{M y}{I}\right] = \frac{(\text{N}\cdot\text{m})(\text{m})}{\text{m}^4} = \frac{\text{N}}{\text{m}^2} = \text{Pa}.\ \checkmark$$

History and Development

• Bernoulli, Euler, Navier. The linear strain assumption is due to Jacob Bernoulli (1690s); Leonhard Euler and Daniel Bernoulli developed the beam equation; and Claude-Louis Navier (1820s) put the flexure formula in its modern engineering form. It is one of the oldest and most reliable results in structural mechanics.

• Wing as a cantilever. Treating the wing as a root-cantilevered beam is the starting point of all wing structural sizing. The spanwise lift distribution sets the shear and bending-moment diagrams; the flexure formula then sizes the spar caps and skins section by section, tapering the structure toward the lightly-loaded tip.

• Deep boxes and the material revolution. Because $I$ rewards depth, wings are built as deep torsion boxes. The shift from aluminium to carbon-fibre composites let designers tailor $I$ and strength directionally, and, combined with active load alleviation, enabled the high-aspect-ratio wings of modern airliners.

Related Concepts: Beam Bending Stress, Section Modulus, Shear Flow, Composite Wing Stiffness, Max Principal Stress, Von Mises Stress 2D, Margin Of Safety

Notes: Registry calculator wing-bending-stress (unverified). The Euler–Bernoulli flexure formula — correct as shipped; duplicates the Mechanical Beam Bending Stress page. Stress is linear in $y$, peaking at the extreme fibre; the neutral axis is centroidal. $Z = I/c$ is the Section Modulus.