Laminate Plate Buckling

Laminate Plate Buckling (structures/composites/laminate-plate-buckling)

Use when you must compute the elastic buckling load of an orthotropic or laminated flat plate under uniaxial in-plane compression from the classical lamination theory bending stiffnesses. This leaf implements the energy-method critical load for a simply supported orthotropic plate, minimizes it over the integer half-wave mode counts, and returns the critical load per unit width, the governing mode and the stability margin against the applied in-plane load, in pure Python, stdlib only. It pairs with structures/fem/plate-buckling for isotropic plates, with structures/composites/laminate-stiffness for the laminate stiffness synthesis that feeds the D terms, and with structures/composites/ sandwich-panels and structures/composites/failure-criteria for the other composite panel limit checks.

Domain quick reference

• Energy-method critical load for mode (m, n), a simply supported orthotropic plate of length a (load direction) and width b with bending stiffnesses D11, D22, D12, D66 in N m: N_x_cr(m, n) = pi^2 * (D11 * (m/a)^2 + 2 * (D12 + 2*D66) * (n/b)^2 + D22 * n^4 * a^2 / (m^2 * b^4)), in N per unit width.
• The D12 + 2*D66 grouping is the torsional contribution and appears squared with the transverse half-wave count n.
• Mode minimization: the governing mode is the integer pair (m, n) in 1..m_max by 1..n_max with the smallest critical load; ties resolve to the smallest (m, n) lexicographically.
• Margin: N_x_cr_min over the applied in-plane compression load per unit width. Below 1.0 the panel buckles under the applied load.
• Isotropic reduction: with D11 = D22 = D and D12 + 2*D66 = D the sweep over a plate long in the load direction reproduces the classic k = 4 result sigma_cr = 4 * pi^2 * D / (b^2 * t).
• Units are SI throughout: N m for the stiffnesses, m for the dimensions, N/m for the critical load, Pa for the stress.
• CMH-17 frames the composite materials context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the panel inputs: the load-direction length a, the width b, and the CLT bending stiffness terms D11, D22, D12 and D66 (N m) of the laminate.
2. Evaluate the energy-method critical load for one candidate half-wave mode (m, n) with critical_load.
3. Sweep the half-wave mode counts, m in 1..m_max and n in 1..n_max, with buckling_mode to minimize the critical load and recover the governing mode pair.
4. Compute the stability margin with buckling_margin, the minimized critical load over the applied in-plane compression load; a margin below 1.0 means the panel buckles.
5. Sanity-check the laminate result with the isotropic reduction: set D11 = D22 = D and D12 + 2*D66 = D and confirm the classic k = 4 stress 4 * pi^2 * D / (b^2 * t) for a plate long in the load direction.
6. Confirm the deterministic contract test offline: python3 scripts/test_laminate_plate_buckling.py.

Worked example

A CFRP skin panel segment under uniaxial compression: a = 0.5 m (load direction), b = 0.25 m, D11 = 200 N m, D22 = 120 N m, D12 = 25 N m, D66 = 45 N m.

• Mode (2, 1): critical_load(200, 120, 25, 45, 0.5, 0.25, 2, 1) returns 86852.5 N/m, that is 86.85 kN/m, matching the prep anchor within 0.1 kN/m.
• Mode sweep: buckling_mode(200, 120, 25, 45, 0.5, 0.25) returns (86852.5, 2, 1): the governing half-wave mode is m = 2, n = 1, not the m = 1 fundamental, because the bending stiffness D11 is high along the load direction.
• Margin: buckling_margin(200, 120, 25, 45, 0.5, 0.25, 40000) returns 2.171 against the applied 40 kN/m; the segment carries 2.17 times the applied load before buckling.
• Sweep truncation: with m_max = 1 the mode is forced to m = 1 and the load jumps to 120014.4 N/m at (1, 1), well above the minimized value, so the full mode sweep matters.
• Isotropic reduction: E = 70 GPa, nu = 0.3, t = 2 mm gives D = 51.28 N m; the sweep over a = 1.0 m, b = 0.25 m returns 32392.5 N/m at mode (4, 1), and dividing by t = 0.002 m gives sigma_cr = 16.196 MPa against the classic 4 * pi^2 * D / (b^2 * t) = 16.196 MPa, within the 16.20 MPa prep anchor.

Verification

• Confirm critical_load(200, 120, 25, 45, 0.5, 0.25, 2, 1) returns 86852.5 N/m and buckling_mode returns (86852.5, 2, 1).
• Confirm m_max = n_max = 2 still captures the (2, 1) mode and that m_max = 1 forces m = 1 with a higher load.
• Confirm the margin 2.171 at 40 kN/m applied and the isotropic reduction stress 16.196 MPa from the minimized N/m divided by t = 0.002 m.
• Confirm monotonicity: the critical load rises with D11 and with a narrower width b.
• Confirm determinism and tie resolution: repeated runs return the same pair, ties resolve to the smallest (m, n).
• Confirm every non-positive D11, D22, D12, D66, a, b, m, n and every non-positive applied load raises ValueError.
• Run the contract test offline: python3 scripts/test_laminate_plate_buckling.py.

Related leaves

• structures/fem/plate-buckling: isotropic plates buckled with a single coefficient k from the edge support and width, no D-matrix path.
• structures/composites/laminate-stiffness: the in-plane laminate stiffness synthesis that precedes the bending terms used here.
• structures/composites/sandwich-panels: face wrinkling into the core, a different composite panel failure mode.
• structures/composites/failure-criteria: lamina failure indices under the in-plane loads.

Pitfalls

• Applying a single isotropic coefficient k to an orthotropic laminate: the mode energy carries the four D terms, and the minimized load over the half-wave sweep differs from any fixed-mode estimate (the m = 1 estimate runs 120.0 kN/m against the 86.85 kN/m minimum here).
• Reporting the fundamental m = 1 mode as the critical mode: high D11 shifts the governing half-wave count above one, as the (2, 1) mode shows in the worked example.
• Dropping the torsional term: 2 * (D12 + 2*D66) * (n/b)^2 contributes 3680 of the 8800 N/m inner sum at mode (2, 1), so neglecting D12 and D66 badly understates the load.
• Confusing load per unit width with stress: the module returns N/m; divide by the panel thickness t to compare with a stress allowables value in MPa.
• Inconsistent units: stiffnesses in N m with dimensions in mm silently scale the result by 10^6; keep SI throughout.
• Reading a margin at or above 1.0 as safe margin: the margin is the ratio, so 2.171 means 2.171 times the applied load, not 2.171 percent of reserve.
• Truncating the mode sweep on a long panel: the governing half-wave count in the load direction grows near a/b, so keep m_max at or above the aspect ratio of the panel.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_laminate_plate_buckling.py

The test covers the worked-example contract (mode (2, 1) at 86.85 kN/m within 0.1 kN/m, minimized load 86852.5 N/m, margin 2.171 within 0.01), the isotropic reduction anchor at 16.20 MPa within 0.05 MPa, the sweep capture at reduced m_max and n_max, the forced m = 1 higher load, monotonicity in D11 and width, determinism with tie resolution to the smallest (m, n), and ValueError rejection of every non-positive stiffness, dimension, mode count and applied load.

Compliance

• Standards referenced, not reproduced: CMH-17 (SAE) is cited as the composite materials context; the buckling relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.