A bending moment is the internal torque at a cross-section of a beam or structural member that results from transverse (perpendicular) loads. When a wing generates lift, the aerodynamic pressure distributed along the span creates a bending moment that is maximum at the wing root and zero at the tip. This root bending moment is the single most important structural load in wing design.
M = F × d
where M is the bending moment, F is the applied force, and d is the distance from the force to the cross-section. For a distributed load (like aerodynamic lift), the bending moment at any spanwise station is the integral of the lift distribution outboard of that station multiplied by its distance.
How bending moment produces stress
A bending moment creates stress that varies linearly across the cross-section: one side is in tension, the other in compression, and a neutral axis between them has zero stress.
σ = M × y / I
where σ is the bending stress at distance y from the neutral axis, M is the bending moment, and I is the second moment of area (moment of inertia) of the cross-section. This equation is the foundation of structural sizing in aerospace: it tells the engineer how much material is needed, and where, to keep bending stresses below the yield strength.
The stress is highest at the surfaces farthest from the neutral axis. This is why efficient beam cross-sections (I-beams, C-channels, hollow tubes) concentrate material at the top and bottom — where the stress is highest — and use thin webs in the middle. A wing spar follows the same logic: thick spar caps at the top and bottom of the airfoil carry bending loads; a thin web between them carries shear.
Bending in aircraft wings
For a wing generating 1g lift on an aircraft of weight W, the total lift per semi-span equals W/2. Assuming an approximately elliptical lift distribution, the root bending moment is roughly:
M_root ≈ (W/2) × (b/2) × 0.42
where b is the total wingspan and 0.42 is the centroid factor for an elliptical distribution. For a 1.5 kg UAV with 1.2 m wingspan at 3g:
M_root ≈ (1.5 × 9.81 × 3 / 2) × (1.2 / 2) × 0.42 ≈ 11.1 N·m
This number drives every structural decision: spar cross-section, infill density at the root, wall thickness, and print orientation (for 3D-printed wings, filament alignment along the span resists bending; alignment across the span does not).
Aspect ratio and bending
High aspect ratio wings produce the same total lift as low-AR wings but distribute it over a longer span. The bending moment at the root increases with span for the same total lift — roughly proportional to b for an elliptical distribution. This is the structural cost of aerodynamic efficiency: a high-AR wing has less induced drag but needs a heavier spar to resist the higher root bending moment. Low-AR delta planforms trade aerodynamic efficiency for structural simplicity precisely because their short spans produce modest bending moments.
Related terms
- Shear Force — the transverse force that accompanies bending moment in loaded beams
- Torsion — the twisting load that accompanies bending in swept and asymmetrically loaded wings
- Stress — the internal force per unit area that bending moment produces
- Structural Load — the external forces that create bending moments