wing-structure

name	wing-structure
description	Aircraft wing structural analysis — box beam idealization (spar caps, skins, ribs), bending moment and shear distribution from aerodynamic loading (parabolic lift distribution, gust loads), shear flow in multi-cell closed sections, buckling of stiffened panels (Euler column, Euler plate), wing deflection and twist, material selection (Al 2024/7075, CFRP), wingbox sizing, CS-25 and FAR Part 25 design loads (limit and ultimate), and fail-safe/damage tolerance.
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Aircraft Wing Structural Analysis — Complete Skill

Wing Loading and Design Conditions

Design Load Cases (CS-25 / FAR Part 25)

Limit load: maximum load expected in service (must carry without permanent deformation) Ultimate load: limit load × 1.5 (must carry without collapse; permanent deformation allowed) Factor of safety: 1.5 (CS-25 25.303; FAR 25.303)

Design load cases:

1. 2.5g pullup maneuver (n_lim = +2.5 for transport; +3.5 or higher for aerobatic)
2. −1.0g pushover (negative limit load factor)
3. Gust load (FAR 25.341): U_de = 56 ft/s (extreme gust at Vc); Dn = 1 ± K_g × U_de × a × V_e / (498 × W/S)
4. Roll maneuver (asymmetric load)
5. Landing gear load (localized)

Critical design: 2.5g pullup for most wings Wing lift = n × W [n = load factor; W = MTOW] Wing root bending moment: M_root = ∫₀^b (n × w(y) − weight_structure × y) × (b/2 − y) dy

Spanwise Load Distribution

Elliptical distribution (ideal): l(y) = l₀ × √(1 − (2y/b)²) [lift per unit span; l₀ at root; b = wingspan; y = spanwise position] l₀ = 4 × L / (π × b) [lift at root for elliptic; L = total lift]

Trapezoidal approximation: l(y) = l_root − (l_root − l_tip) × (2y/b) [linear taper from root to tip]

Bending moment distribution: M(y) = ∫_y^(b/2) (l(η) − n_gear(η) − n_engine(η)) × (η − y) dη Integrate from tip to section y; subtract engine and gear weight contributions

Root bending moment (elliptical lift, parabolic weight): M_root ≈ L × b / (4π) × (n × 0.637 − structure_fraction × 0.25) [simplified] Or numerically from trapezoid integration at each station

Shear Force Distribution

Shear V(y) = dM/dy: V(y) = ∫_y^(b/2) (l(η) − n_fuel(η) − n_structure(η)) dη Wing fuel acts as relief loading (reduces bending moment) Critical: minimum fuel case (no fuel weight relief) for maximum M_root

Wingbox Cross-Section

Idealized Box Beam

Components: Front spar: carries most shear; located ~15–25% chord Rear spar: secondary shear; ~60–70% chord Upper/lower skins: primarily carry bending (axial) loads (tension lower; compression upper at +g) Ribs: maintain cross-section shape; transfer loads between skins and spars; bay structure

Boom idealization: Skins and spars idealized as shear panels (no bending stiffness) Spar caps (booms) carry all axial (bending) load: σ = M × z / I [z = distance from neutral axis] Effective boom area: A_eff = A_cap + (t_skin × s) / 6 [s = panel width; t = skin thickness; for tapered panels different formula]

Bending Stress

Bending stress in spar cap: σ = M_x × z / I_xx [z = height above/below neutral axis; I_xx = second moment of area about horizontal axis] I_xx ≈ 2 × A_cap × h_spar² / 4 + I_skin [for simplified boom section; h = spar height]

For two booms (top and bottom cap): I_xx = A_upper × z_upper² + A_lower × z_lower² [if symmetric: I = 2 × A × (h/2)²] σ_cap = M × (h/2) / I [stress in top/bottom cap; upper = compression at positive g]

Strength check: σ_ultimate = M_ult × c / I ≤ σ_allow = F_tu / safety [F_tu = ultimate tensile strength; σ_allow usually σ_cy for compression]

Shear Flow Analysis

Shear flow in closed section (thin-wall): q(s) = −V_z × Q(s) / I_xx + q₀ [Q(s) = first moment of area; q₀ = shear flow constant from torsion]

Torsion shear flow (Bredt-Batho): q_torsion = T / (2 × A_enclosed) [T = torque; A_enclosed = area enclosed by section] Angle of twist: dθ/dx = T / (4 × G × A_enclosed² / Σ(q × ds/t))

Multi-cell wingbox: Two-cell box (with middle spar): solve for q₀ in each cell using compatibility (equal twist rate) Compatibility: ∮(q/Gt) ds = same for each cell if shear center coincides with aerodynamic center

Panel Buckling

Skin-Stiffener Panels (Compression)

Flat unstiffened plate buckling: σ_cr = K_c × π² × E / (12 × (1−ν²)) × (t/b)² [K_c = buckling coefficient; t = skin thickness; b = stiffener spacing] K_c depends on boundary conditions and a/b ratio: Simply supported all sides: K_c = 4 (b/a < 1); K_c → higher for longer panel One edge free: K_c = 0.425 (flange/stringer outstand)

Post-buckling diagonal tension field: Stiffened panels can carry load beyond initial buckling via diagonal tension Effective width method: b_eff = 2t × √(E / σ_applied) [portion of skin still contributing after buckling] For efficient design: design for buckling at limit load; ultimate load via diagonal tension

Stiffener column buckling (Euler): P_cr = π² × E × I_stiffener / (K × L_e)² [K = end condition factor; L_e = effective length = rib pitch] Stiffener cross-section must satisfy: P_cr > σ_applied × (A_stiffener + b_eff × t_skin)

Minimum Skin Gauge

Shear buckling of panels (spar web, skin under shear): τ_cr = K_s × π² × E / (12(1−ν²)) × (t/b)² [K_s = 5.34 + 4(b/a)² for long panels] Design: τ_cr ≥ τ_applied = V/(A_web) for no shear buckling at limit load

Minimum skin thickness for fuel tank sealing, access panels: typically 1.0–1.5 mm (Al); 0.8–1.2 mm (CFRP laminate)

Wing Deflection and Twist

Tip Deflection

Tip deflection (semi-span b/2): δ_tip = ∫₀^(b/2) M(y) × y / (E × I(y)) dy [numerical integration due to variable I(y)] Simplified (uniform EI, parabolic M): δ_tip ≈ M_root × (b/2)² / (3 × E × I_root)

Typical deflection: Transport aircraft: δ_tip / (b/2) = 5–10% at limit load (Boeing 787: ~4 m tip deflection at MTOW 2.5g) Flexible wing → aeroelastic effects important

Tip twist (aileron reversal concern): θ_tip = ∫₀^(b/2) T(y) / (G × J(y)) dy [T = torsion; J = torsion constant] Aileron reversal: at V_reversal = 2 × G × J × e_a / (q × c² × a) [e_a = elastic axis offset from aero center] V_reversal must be > V_D (dive speed) for structural stiffness adequacy

Material Selection

Aluminum Alloys (Conventional)

2024-T3 (Lower surface skins, tensile structure): F_tu = 483 MPa; F_ty = 345 MPa; F_cy = 324 MPa; K_Ic = 44 MPa√m Excellent damage tolerance (slow crack growth); preferred for tension-dominated lower surface

7075-T6 / 7075-T73 (Upper surface skins, compression): F_tu = 572 MPa; F_ty = 503 MPa; F_cy = 476 MPa; K_Ic = 27 MPa√m (T6) / 35 MPa√m (T73) T73: lower strength but better SCC resistance; preferred for thick sections Upper surface: compression-dominated → want high F_cy and buckling resistance

7150-T77 / 7055-T7751 (Modern alloys): Higher strength than 7075-T73 with improved toughness; 7-series with secondary alloying (Cu, Zr) Boeing 777, 787 fuselage frames

CFRP (Modern Aircraft)

Aerospace CFRP layup: IM7/977-3 or T800/M21 prepreg; [0/±45/90] quasi-isotropic: E = 54 GPa; σ_ult = 600 MPa (tension) [0-dominated]: E = 105 GPa; σ_ult = 900 MPa; for spar caps Compression strength lower: σ_cy = 500–700 MPa (CAI, compression after impact)

Damage tolerance (CS-25 25.571): Fail-safe: loss of any single load path must not cause catastrophic failure Damage tolerant: structure with initial crack grows slowly; inspection interval < crack growth life CFRP: no crack propagation (different from metals); delamination is primary damage mode Barely Visible Impact Damage (BVID): barely visible at 2 J impact → 15% compression strength knockdown

CS-25 Requirements Summary

CS-25.303: factor of safety = 1.5 for all limit loads CS-25.305: strength and deformation (no permanent deformation at limit; no failure at ultimate) CS-25.341: gust loads (discrete gust; continuous turbulence) CS-25.571: damage tolerance and fatigue evaluation CS-25.629: flutter requirements (no flutter up to 1.15 × VD at any altitude)

Standards and References

Standard	Scope
CS-25	EASA certification standards for large aeroplanes
FAR Part 25	FAA equivalent to CS-25
MIL-HDBK-5J / MMPDS	Material properties for aerospace
CMH-17 (Composite Materials Handbook)	CFRP material and design data
Bruhn "Analysis and Design of Aircraft Structures"	Classic aircraft structures reference
Megson "Aircraft Structures for Engineering Students"	University reference

Output

Provide: wing geometry (span b [m]; aspect ratio AR; taper ratio λ; sweep angle [°]; root chord c_r [m]; tip chord c_t [m]; t/c at root [%]), design loads (MTOW [kg]; design load factor n_ult = 1.5 × n_lim; MTOW × g × n_ult = design lift [kN]; gust load if applicable), spanwise load distribution (trapezoid or elliptic; lift distribution l(y); engine/fuel weight relief [kN/m]; net distributed load), section properties (wingbox dimensions at root: h [mm]; width [mm]; A_cap [mm²] top and bottom; skin thickness t [mm]; I_xx [cm⁴]; A_web [cm²]), bending stress (M_root [kN·m]; σ_cap = M×(h/2)/I [MPa]; vs. F_cy(material) [MPa]; margin [%]), shear flow (V_root [kN]; q_max in skins and spar webs [N/mm]; q_torsion [N/mm]; τ_max [MPa]), panel buckling (σ_cr for upper skin panel [MPa]; vs. σ_applied [MPa]; post-buckling reserve; stiffener sizing), tip deflection (δ_tip [m] at n_lim load; δ_tip/(b/2) [%]; tip twist [°]), material (spar caps: 7075-T73 or UD CFRP; skins: 2024-T3 or ±45 CFRP; wing ribs: 2024-T3), damage tolerance (fail-safe provision; inspection interval; BVID knockdown if CFRP), and applicable standard (CS-25 25.303/25.305/25.341/25.571; MMPDS for Al properties; CMH-17 for CFRP).