| name | plate-buckling |
| description | Plate buckling — Euler plate buckling (SSSS, various BCs), effective width, post-buckling, stiffened panels, AISC/API/aerospace criteria, buckling coefficients k. |
| metadata | {"priority":7,"promptSignals":{"phrases":["plate buckling","thin plate buckling","effective width","stiffened plate","web buckling","shear buckling"],"minScore":3}} |
Plate Buckling — Complete Skill
Critical Buckling Stress (Timoshenko)
General formula:
σ_cr = k × π²E / (12(1-ν²)) × (t/b)²
k = buckling coefficient (depends on aspect ratio a/b, load type, boundary conditions)
t = plate thickness; b = shorter plate dimension (width under compression)
E = elastic modulus; ν = Poisson's ratio (0.3 for steel)
Buckling Coefficients k
Uniaxial Compression
| Boundary Conditions | k (a/b ≥ 1) |
|---|
| SSSS (all sides simply supported) | 4.0 |
| SSSF (3 sides SS, one free) | 0.425 (free edge parallel to load) |
| SSSS, free edge long side | 1.28 (one unloaded edge free) |
| CCCC (all clamped) | 6.97 |
| CSSS | ~5.42 |
SSSS (most common design case): k = 4.0 for square plate, a/b ≥ 1
Shear Buckling
| Boundary Conditions | k_s |
|---|
| SSSS | 5.34 + 4(b/a)² (long plate) |
| CCCC | 8.98 + 5.6(b/a)² |
Shear buckling stress: τ_cr = k_s × π²E / (12(1-ν²)) × (t/b)²
Combined Compression + Shear (Interaction)
R_σ² + R_τ² = 1 (SSSS plate, conservative)
R_σ = σ/σ_cr; R_τ = τ/τ_cr
Effective Width (Post-Buckling)
Plates have significant post-buckling reserve:
Effective width concept (von Kármán / Winter):
b_eff = b √(σ_cr/σ) or b_eff = b [1 - 0.22/λ_p] / λ_p
Plate slenderness: λ_p = √(f_y/σ_cr) = (b/t) / (28.4 ε √k_σ) (Eurocode notation)
ε = √(235/f_y); f_y in MPa
For stocky plates (λ_p ≤ 0.673): b_eff = b (no reduction)
For slender (λ_p > 0.673): b_eff/b = (1 - 0.22/λ_p)/λ_p ≤ 1.0 (Winter formula)
AISC Plate Classification
b/t limits for local buckling (AISC 360 Table B4.1):
Compact (no buckling before yield):
- Unstiffened element: b/t ≤ 0.56√(E/F_y) [plates, flanges]
- Stiffened element (web): h/t_w ≤ 2.24√(E/F_y) for shear
Noncompact: reduced capacity; slender: buckling governs
Web Shear Buckling (AISC G2)
Without tension field:
V_n = 0.6 F_y A_w C_v1
C_v1 = 1.0 for h/t_w ≤ 2.24√(E/F_y)
C_v1 from √(kv E/F_y) / (h/t_w) × 1.10 to 1.37 for intermediate
With tension field action (stocky webs, AISC G3):
V_n = 0.6 F_y A_w [C_v2 + (1-C_v2)/√(1.15√(1+(a/h)²))]
(Post-buckling tension field contribution)
Stiffened Panels
Longitudinal stiffeners: increase σ_cr by raising equivalent k
Effective k for stiffened plate depends on stiffener EI_s relative to plate bending stiffness D
Reference: NACA TN 733 (Timoshenko-Gere)
Orthotropic plate theory:
D_x = E t_x³/(12(1-ν²)); D_y = E t_y³/... for variable thickness sections
Buckling of stiffened panel: use smeared properties or folded-plate method
Buckling Interaction (Combined Loads)
Compression N + bending M + shear V:
AISC: R_n² + R_v² ≤ 1 (web combined shear + bending)
Eurocode EN 1993-1-5: η₁ + η₃² ≤ 1 (bending + shear interaction)
Aerospace Criteria (NACA/NASA)
NASA SP-8007, MIL-HDBK-5 (MMPDS):
Reserve factor = allowable / applied
Margin of safety MS = RF - 1 ≥ 0 (FAR 25 requires positive MS at limit load)
Interaction equations for combined buckling modes per NACA TN-3781
Design Procedure
- Identify load type (compression, shear, combined) and boundary conditions
- Calculate b/t ratio; classify as compact/noncompact/slender
- Compute σ_cr using appropriate k
- If slender: apply effective width to get reduced area
- Check combined loading interaction
- If web: check AISC G2/G3 for shear; add transverse stiffeners if needed
- Verify MS ≥ 0 (aerospace) or φσ_cr ≥ σ_demand (LRFD)
Output
Provide: b/t ratio, k value, σ_cr [MPa], λ_p (plate slenderness), b_eff/b, applied stress ratio σ/σ_cr, interaction ratio (if combined), MS or safety factor, stiffener recommendation if needed.
