Aerodynamic and physical foundations of Multi-Cell Wing Box Torsion.
Modern aircraft wings are constructed around stressed-skin semi-monocoque wing boxes comprising upper and lower aerodynamic skins, front and rear main spars, and intermediate web partitions, forming a closed multi-cell thin-walled structural tube. Under aerodynamic flight loads, the wing box resists massive torsional moments produced by center-of-pressure offsets and control surface deflections. For closed thin-walled tubes, Saint-Venant's warping theory is formulated through Bredt-Batho shear flow theory. In a single closed cell of enclosed cross-sectional area A_m, shear stress tau is uniformly distributed across wall thickness t as shear flow q = tau * t, satisfying the first Bredt-Batho relation: T = 2 * A_m * q. In a multi-cell wing box with N closed cells, torsional twist rate dtheta/dz must be identical across all interconnected cells to maintain geometric structural compatibility: dtheta/dz = (1 / (2 * G * A_i)) * oint_i (q / t) ds. The total applied torque equals the sum of cell torque contributions: T = sum (2 * A_i * q_i). The effective torsional rigidity is G * J = T / (dtheta/dz). In this laboratory experiment, a 2-cell aircraft wing box model is subjected to pure torsional torque from T = 0.6 to 3.0 kN-m. Measuring the torsional twist rate dtheta/dz validates the linear compliance 10^3 / (G * J) = 1.1765 mrad/(kN-m^2).
Viva voce questions & analytical theory
What is the first Bredt-Batho relation for a single closed thin-walled cell in torsion?
The first Bredt-Batho relation relates the applied torque T to the circulating shear flow q and enclosed cell area A_m: T = 2 * A_m * q. It proves that for a closed cell, shear flow is constant around the perimeter regardless of wall thickness variations.
How are shear flows determined in a multi-cell wing box?
In a multi-cell structure, the rate of twist dtheta/dz must be identical for every cell to maintain geometric continuity. For N cells, N compatibility equations dtheta/dz = (1 / (2*G*A_i)) * oint_i (q/t) ds are formulated, and the sum of cell torques equals applied torque: T = sum(2 * A_i * q_i). Solving this linear system yields the shear flow in every wall.
What happens to the shear flow in a web shared between two adjacent cells?
In an interior web dividing cell 1 and cell 2, the net shear flow is the algebraic difference between the circulating shear flows of the two cells: q_web = q_1 - q_2. If the shear flows in both cells circulate in the same direction, they oppose each other in the common web, reducing web shear stress.
Why does a closed wing box have dramatically higher torsional rigidity than an open section of identical weight?
A closed box allows shear flow to circulate continuously around a large enclosed area A_m. By Bredt-Batho theory, torsional stiffness scales with the square of the enclosed area: J = 4 * A_m^2 / oint (ds/t). An open section has zero enclosed area, and its stiffness scales only with wall thickness cubed: J = (1/3) * sum(s_i * t_i^3).
What is diagonal tension in thin aircraft wing skins after shear buckling?
Herbert Wagner demonstrated that when a thin metallic web buckles under shear stress, it does not collapse. Instead, it buckles into diagonal wrinkles that act like tension diagonals in a Pratt truss. The skin carries additional shear through pure diagonal tension, allowing aircraft structures to operate safely beyond their initial buckling load.
How do internal wing ribs contribute to the torsional efficiency of a wing box?
Wing ribs maintain the aerodynamic airfoil profile, prevent chordwise crushing, and provide rigid cross-sectional shape stability that suppresses cross-sectional distortion. Furthermore, ribs act as bulkheads that introduce concentrated torques into the skin as circulating shear flows.
Experimental precautions and wind tunnel safety
- Ensure the applied hydraulic forces form a perfectly balanced couple to avoid introducing transverse shear or bending moments.
- Verify that skin shear stresses do not approach the critical shear buckling stress of thin unstiffened skin panels.
- Check that all interior web rosette strain gauges are temperature-compensated to eliminate thermal drift.
- Do not exceed the elastic shear limit of the riveting joints during high-torque testing.
Theoretical assumptions & flow limits
- Bredt-Batho theory assumes constant shear flow along each segment without localized rivet hole stress concentrations.
- Root warping restraint increases torsional stiffness in the immediate vicinity of the root clamp (z < 1.5 * chord).
- Assumes thin-walled skin behavior without skin panel post-buckling diagonal tension fields.
- Internal rib flexibility can cause minor cross-sectional shape distortion (anti-clastic curvature).