Plate Buckling (structures/fem/plate-buckling)

Use when the task is the elastic stability of a flat plate or skin panel: resolving the plate buckling coefficient k from the edge conditions and the panel aspect ratio, computing the critical compression buckling stress sigma_cr = k * pi^2 * E / (12 * (1 - nu^2)) * (t / b)^2 and the shear buckling stress tau_cr of a web or panel, running the combined compression-shear interaction check, and sizing the effective width of stiffened skin in the post-buckling range. The logic module is pure Python standard library (no numpy, no FEA software) and deterministic. Units are SI: E in Pa, t and b in m, stresses in Pa, a and b in m.

Domain quick reference

• A flat rectangular plate of thickness t and width b (measured across the load direction) buckles elastically when the applied edge stress reaches the critical value:

  sigma_cr = k * pi^2 * E / (12 * (1 - nu^2)) * (t / b)^2
  tau_cr  = k_s * pi^2 * E / (12 * (1 - nu^2)) * (t / b)^2
  

  where E is the Young's modulus, nu the Poisson ratio, and k (or k_s) the plate buckling coefficient. The stress scales with the square of the thickness-to-width ratio: doubling t / b quadruples the critical stress, which is why skin thickness and stringer pitch trade directly against each other.

• Compression coefficient, simply supported on all edges (exact, minimized over the half-wave count m):

  k = min_m (m / a_r + a_r / m)^2,   a_r = a / b
  

  The long plate gives k = 4.0. The clamped long plate approximation is k = 6.97; short clamped plates have a higher coefficient and need tabulated data. Shear coefficient (Timoshenko):

  simply supported:  k_s = 5.34 + 4 / a_r^2    (a_r >= 1)
                     k_s = 5.34 * a_r^2 + 4    (a_r <  1)
  clamped:           k_s = 8.98 + 5.6 / a_r^2  (a_r >= 1)
                     k_s = 8.98 * a_r^2 + 5.6  (a_r <  1)
  

• Combined compression and shear interacts approximately as

  sigma / sigma_cr + (tau / tau_cr)^2 <= 1
  

  linear in compression and quadratic in shear: a modest shear stress consumes a large share of the buckling capacity.

• Post-buckling: stiffened skin carries load through the von Karman effective width b_e = 1.9 * t * sqrt(E / sigma_edge), capped at the panel width, valid once the edge stress exceeds the panel buckling stress.

Worked anchors (verified by running scripts/plate_buckling_logic.py): an aluminum skin with E = 70 GPa, nu = 0.33, t = 2 mm, stringer pitch b = 150 mm and a/b = 2, simply supported, has k = 4.0 (the m = 2 half-wave gives (1 + 1)^2) and sigma_cr = 45.9 MPa; against an applied compression of 30 MPa the margin of safety is 0.531. An aluminum spar web with t = 1.5 mm, depth b = 250 mm and a/b = 2 has k_s = 5.34 + 4/4 = 6.34 and tau_cr = 14.7 MPa; against 8 MPa of applied shear the margin is 0.844. With 30 MPa compression and 8 MPa shear the interaction index is 0.947, stable with margin 0.056. At an edge stress of 200 MPa the 2 mm skin has an effective width of 71.1 mm. Second anchor: the same skin at a/b = 1.5 gives k = 4.340 and at a/b = 0.5 gives k = 6.25, while the clamped long plate gives k = 6.97, 74 percent above the simply supported value.

Workflow

1. Identify the panel geometry: the loaded length a, the width b across the load (for a skin panel between stiffeners this is the stiffener pitch, for a spar web the web depth), and the thickness t. Compute the aspect ratio a_r = a / b.
2. Select the edge condition: 'ssss' for a simply supported panel (skin between stringers with flexible attachments) or 'cccc' for a clamped panel (heavily restrained edges). Aliases accepted: simply-supported, pinned, clamped, fixed.
3. Resolve the coefficient with compression_coefficient(a_r, edge_condition) for compression or shear_coefficient(a_r, edge_condition) for shear.
4. Compute the critical stress with compression_buckling_stress(E, nu, t, b, k) or shear_buckling_stress(E, nu, t, b, k_s). Both share the same denominator 12 * (1 - nu^2).
5. Run the complete margin check in one call with compression_panel_check(E, nu, t, a, b, edge_condition, applied_stress) or shear_panel_check(E, nu, t, a, b, edge_condition, applied_shear), which return the coefficient, the critical stress, the margin of safety and the stable verdict. Apply the required factor of safety from the certification basis (1.5 ultimate-to-limit per FAR-25.303 / CS-25.303) before comparing the applied stress against the critical stress.
6. For combined compression and shear (a shear web carrying bending compression plus shear, or a skin panel under shear plus compression), run interaction_index(compression_stress, compression_critical, shear_stress, shear_critical): stable when the index is below 1, margin = 1 / index - 1.
7. For stiffened skin loaded beyond its buckling stress, size the effective width with effective_width(E, sigma_edge, t) and cap it at the panel width b before re-computing the stiffener load.

Pitfalls

• Routing column buckling here: buckling-analysis handles 1D columns and struts with the Euler load, effective length factor K and slenderness ratio; plate-buckling handles 2D flat plates and skin panels with the k-coefficient formulas and never uses a slenderness ratio or an effective length factor.
• Routing FEA buckling here: a full-model eigenvalue buckling run in CalculiX belongs to calculix-linear or calculix-nonlinear; plate-buckling is a hand-scale closed-form panel check with no stiffness matrix and no software.
• Routing panel sizing here: fuselage-skin-stringer and wing-box-sizing (vehicle-design family) close the overall stiffened shell or wing box (skin thickness from hoop stress, stringer area, spar cap area); plate-buckling only checks the elastic stability of one flat panel with known dimensions.
• Routing sandwich panels here: sandwich-panels checks face stress, core shear and face wrinkling of a sandwich construction; plate-buckling checks monolithic flat skins and webs for elastic instability, not a sandwich cross-section.
• Using the column coefficient for a plate: k = 4.0 (simply supported) or 6.97 (clamped) applies to flat plates; the Euler column result is a different geometry and a different formula.
• Mixing the aspect ratio convention: a is the loaded length and b the width across the load; swapping them changes k (a/b = 0.5 gives k = 6.25, a/b = 2 gives k = 4.0) and the critical stress.
• Forgetting that clamped short plates are special: the 6.97 value is the long-plate clamped approximation; compression_coefficient raises ValueError below aspect_ratio 1 where tabulated data is required, so do not silently extrapolate.
• Applying the interaction equation wrongly: the compression term is linear and the shear term is squared; using (tau / tau_cr) to the first power overstates the margin.
• Using effective width below the buckling stress: the von Karman width is a post-buckling concept and must be capped at the panel width; below the critical stress the full width carries load.
• Mixing units: E in GPa with t and b in mm silently corrupts the stress by factors of 1e9 or 1e6; keep everything SI (Pa, m).
• Ignoring imperfections: the ideal flat plate formulas assume a perfect plate; initial waviness and eccentric load reduce the real buckling stress below sigma_cr, and in-service corrosion or dent damage lowers the effective thickness.

Behavior contract (gate 3)

The plate buckling logic is exercised by the gate 3 contract test: scripts/test_plate_buckling.py against scripts/plate_buckling_logic.py (stdlib unittest, offline). It asserts the worked anchors above, the coefficient boundaries at aspect ratios 1 and 2, the (t/b)^2 and E scalings, the clamped versus simply supported ranking, the linear-compression and quadratic-shear interaction scaling, the effective width dependence, and the ValueError cases for non-positive, non-finite or unknown inputs. Run:

python3 scripts/test_plate_buckling.py

Compliance

• FAR-25 and CS-25 are referenced, not reproduced: standards-map.yaml marks them gated: false and reference-only: true; only the summary paraphrase above is used, never standard text.
• compliance: STANDARDS-REF, gated: false.