Buckling of thin plates and stiffened panels

Why it matters

Aircraft skins are so thin that, in compression or shear, they buckle long before the material yields. The upper wing skin in a positive-g manoeuvre, the fuselage crown under sagging, and every spar web in shear are designed around buckling, not strength. Stringers and ribs exist largely to raise buckling stresses, and the design of a stiffened panel is a contest between skin buckling, stringer column buckling and material failure.

Key ideas

Plate buckling. A flat plate under in-plane compression stays flat until the compressive stress reaches a critical value, then deflects out of plane in a wave pattern. It is a stability problem: the critical stress depends on stiffness (E, ν), geometry (t/b) and edge support, not on strength.

Critical stress. For a rectangular plate of width b (loaded edges a apart) and thickness t, σ_cr = k·π²·E/(12(1 − ν²))·(t/b)². The buckling coefficient k depends on the edge conditions, the aspect ratio a/b and the type of load. For a simply supported plate in uniaxial compression buckling into m half-waves along its length, k = (m·b/a + a/(m·b))². The minimum over m is k = 4.0, reached whenever a/b is an integer; for long plates (a/b > about 3) k ≈ 4 regardless of length, so the width b controls the stress. A square plate buckles in one half-wave and a plate with a/b = 2 in two.

Typical k values (take precise ones from data sheets such as ESDU or Bruhn):

• All edges simply supported, uniaxial compression, long plate: k = 4.0.
• Unloaded edges clamped, long plate: k ≈ 6.98.
• One unloaded edge simply supported, the other free (a stringer flange): k ≈ 0.43.
• Shear, all edges simply supported, long plate: k_s ≈ 5.35 (9.34 for a square plate).

Inelastic buckling. If σ_cr from the elastic formula exceeds the proportional limit, replace E by a tangent or reduced modulus (from the material's stress–strain curve) or use a cut-off at yield.

Post-buckling and effective width. A skin panel supported by stringers does not collapse when it buckles: the material near the stringers stays nearly flat and keeps carrying load, while the centre of the panel sheds load. This is modelled by an effective width b_e of skin acting with each stringer at the stringer stress σ_s. Von Kármán's estimate is b_e = 1.9·t·√(E/σ_s) (total, half on each side of the stringer); it cannot exceed the actual stringer pitch.

Stiffened panel failure modes.

• Local skin buckling between stringers (initial buckling, usually allowed below limit load in compression skins only if fatigue and stiffness are acceptable).
• Local buckling or crippling of the stringer section itself (flanges with k ≈ 0.43).
• Column (flexural) buckling of the stringer with its effective skin between ribs: σ = π²·E·c/(L/ρ)², where ρ is the radius of gyration of stringer plus effective skin, L the rib spacing and c an end-fixity coefficient (c = 1 pinned, up to 4 fixed).
• Flexural-torsional buckling of open stringers, inter-rivet buckling of skin between fasteners, and wrinkling of sandwich faces. A good design makes these critical stresses comparable, so no single mode wastes weight.

Shear buckling. Spar webs and skin panels in shear buckle into diagonal waves at τ_cr = k_s·π²·E/(12(1 − ν²))·(t/b)², b being the shorter side. Beyond this, thin webs can carry load as a tension field (next topic).

Formulas

σ_cr = k·π²·E / (12·(1 − ν²))·(t/b)²

• σ_cr: critical compressive stress (Pa), k: buckling coefficient, E: Young's modulus (Pa), ν: Poisson's ratio, t: thickness (m), b: loaded-edge width (m).

k = (m·b/a + a/(m·b))²

• Simply supported plate, uniaxial compression; a: length in the load direction (m), m: number of half-waves. Choose the integer m giving the smallest k.

D = E·t³ / (12·(1 − ν²)) and N_cr = k·π²·D / b²

• D: flexural rigidity (N·m), N_cr: critical load per unit width (N/m) = σ_cr·t.

τ_cr = k_s·π²·E / (12·(1 − ν²))·(t/b)²

• Shear buckling stress (Pa); b: shorter side (m), k_s from data sheets.

b_e = 1.9·t·√(E / σ_s)

• Total effective skin width (m) acting with a stringer at stress σ_s (Pa); b_e ≤ stringer pitch.

σ_col = c·π²·E / (L/ρ)²

• Stringer column buckling. L: rib spacing (m), ρ: radius of gyration of stringer plus effective skin (m), c: end-fixity coefficient.

Worked examples

Example 1 (standard): skin panel in compression. Given: aluminium skin panel, a = 600 mm (load direction), b = 150 mm, t = 2.0 mm, E = 70 GPa, ν = 0.3, all edges simply supported.

1. Aspect ratio a/b = 4, an integer, so m = 4 half-waves and k = (4 × 0.25 + 4/4)² = 4.0.
2. π²·E/(12(1 − ν²)) = 9.870 × 70 000/(12 × 0.91) = 63 266 N/mm².
3. (t/b)² = (2/150)² = 1.778 × 10⁻⁴.
4. σ_cr = 4.0 × 63 266 × 1.778 × 10⁻⁴ = 45.0 N/mm². Answer: σ_cr = 45.0 MPa, far below yield, so the panel buckles elastically.

Example 2 (GATE level): stiffened panel. Given: skin t = 1.5 mm, stringer pitch b = 120 mm, stringer area A_s = 200 mm², rib spacing L = 600 mm (pinned ends, c = 1), radius of gyration of stringer plus effective skin ρ = 12 mm, E = 70 GPa, ν = 0.3.

1. Skin initial buckling (k = 4): σ_cr = 4 × 63 266 × (1.5/120)² = 39.5 N/mm².
2. Stringer column buckling: L/ρ = 600/12 = 50; σ_col = π² × 70 000/50² = 276.3 N/mm².
3. Effective skin width at that stress: b_e = 1.9 × 1.5 × √(70 000/276.3) = 2.85 × 15.92 = 45.4 mm (less than the 120 mm pitch, so valid).
4. Failure load per stringer bay: P = σ_col·(A_s + b_e·t) = 276.3 × (200 + 45.4 × 1.5) = 276.3 × 268.1 = 74.1 kN. Answer: skin buckles at 39.5 MPa; panel fails by stringer column buckling at 276 MPa, about 74 kN per stringer. The ratio shows how much load the buckled skin and stringers carry beyond initial buckling. Check σ_col against the alloy's proportional limit before relying on the elastic value.

Common mistakes

• Using the plate length a instead of the loaded-edge width b in (t/b)².
• Taking k = 4 for a free-edged flange; it is about 0.43.
• Choosing m = 1 for a long plate; the plate buckles in the number of half-waves that minimises k.
• Using the Euler column formula (no Poisson factor) for a plate, or the plate formula for a column.
• Treating initial skin buckling as panel failure; stiffened panels carry much more load in post-buckling.
• Ignoring plasticity when σ_cr comes out above the proportional limit.

For GATE AE

Expect σ_cr of a simply supported plate, the effect of aspect ratio and number of half-waves, the dependence on (t/b)², shear buckling of webs, and effective width. Column buckling of stringers with different end conditions also appears. Practise k = (mb/a + a/mb)² and finding the critical m quickly.

Quick check

1. How does σ_cr change if the skin thickness is doubled?
2. A simply supported plate has a/b = 3. How many half-waves does it buckle into, and what is k?
3. What is k for a long plate with one free unloaded edge?
4. Does the effective width increase or decrease as stringer stress rises? Answers: 1. it becomes four times larger; 2. three half-waves, k = 4; 3. about 0.43; 4. it decreases (b_e ∝ 1/√σ_s).