Formula & Calculator
Wing Root Bending Moment (Elliptical Load, Simplified)
Simplified estimate of bending moment at the wing root assuming an elliptical spanwise lift distribution.
Interpretation
Wing root bending moment: M_root ≈ (4/(3π))·L·(b/2), for elliptical lift distribution. It gives the maximum bending moment at the wing root. Example: L=50,000 N, b=20 m → M_root ≈ (4/(3π))×50000×10 ≈ 212,206 N·m.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| M_root | Root bending moment | N*m |
| L | Total lift force | N |
| b | Wingspan | m |
What it means
The wing root bending moment is a critical load for structural design. For an elliptical lift distribution (minimum induced drag), the bending moment is approximately 4/(3π) times the total lift times the semispan. This approximates the moment from an idealised lift distribution. In reality, the distribution may be different, affecting the bending moment. The formula is used in preliminary structural sizing to estimate the required spar strength. Understanding this relation is important for wing design and for weight estimation.
Worked example
Wing Root Bending Moment – Two Examples
Real‑World| Parameter | Value |
|---|---|
| L | 80,000 N |
| b | 10 m |
| Parameter | Value |
|---|---|
| L | 500,000 |
| b | 35 |
Common mistakes
- Wing root bending moment (elliptical load, simplified): M_root ≈ (4/(3π)) · L · (b/2).
- L: Total lift (N).
- b: Wing span (m).
- Assumes elliptical lift distribution (minimum induced drag).
- Factor (4/(3π)) ≈ 0.424 – for other distributions, use different factor.
Applications
The wing root bending moment, M_root, is a critical structural load that determines the sizing of wing spars. For an elliptical lift distribution, the simplified formula yields M_root ≈ (4/(3π))·L·(b/2). Engineers use this to size the wing structure, to compute bending stresses, and to design the wing‑fuselage attachment. Accurate bending moment is essential for ensuring that the wing can withstand aerodynamic loads without failure. By understanding bending moments, aerospace engineers can design efficient, lightweight wing structures that meet safety and performance goals.
- Wing spar and rib design
- Structural load analysis for certification
- Weight estimation and structural optimisation
- Wing‑fuselage joint design
- Fatigue and damage tolerance assessment
Frequently Asked Questions
It is a simplified estimate of the bending moment at the wing root assuming an elliptical spanwise lift distribution. It is used for preliminary structural sizing.
L = total lift (N)
b = wing span (m)
The factor (4/(3π)) comes from integrating the elliptical load distribution.
It determines the required spar strength and weight. The wing root bending moment is a critical load case for structural design.
Elliptical lift distribution, no sweep, and the lift acts at the quarter‑chord. It is only for preliminary estimation.
- Using a rectangular (uniform) lift distribution assumption when elliptical loading gives a meaningfully different result.
- Neglecting the weight of the wing itself.
- Using the total lift instead of the lift per panel.
For L = 150,000 N, b = 20 m, Mroot ≈ (4/(3π)) × 150000 × (20/2) = 0.4244 × 150000 × 10 = 0.4244 × 1,500,000 = 636,600 N·m.
A more rectangular distribution (higher loading at the tips) increases the root bending moment for the same total lift, making the wing heavier.
Sweep changes the lift distribution and the effective moment arm, requiring a more detailed analysis.
Fuel weight reduces the net bending moment because it acts as a lifting force; the formula above is for aerodynamic lift only.
A higher bending moment requires heavier spars and ribs, increasing the structural weight fraction.