Thermal Buckling

Thermal Buckling (structures/thermal-structures/thermal-buckling)

Use when the task is the thermal buckling of a restrained aerospace structure: a skin panel or column whose free expansion is blocked, so a temperature rise builds an in-plane compressive load that can buckle the member at a critical temperature. This leaf computes the elastic buckling stress of a flat plate under uniform compression, the compressive stress developed by a constrained temperature change under uniaxial or biaxial restraint, the critical temperature rise that drives a plate or an Euler column to buckling, and the resulting margin. The logic module is pure Python standard library (no numpy, no FEA software) and deterministic. Units are SI: E in Pa, alpha in 1/K, thickness, width and lengths in m, stresses in Pa, temperature rise in K. It pairs with structures/thermal-structures/thermal-stress-analysis for fully constrained and bimetallic members and with structures/fem/plate-buckling for mechanically loaded plates.

Domain quick reference

• Plate flexural rigidity:

  D = E * t**3 / (12 * (1 - nu**2))
  

• Elastic buckling stress of a flat plate under uniform compression (long simply supported plate):

  sigma_cr = k * pi**2 * D / (b**2 * t)
  

  with b the loaded-width direction dimension and the edge condition coefficient k = 4.0 for a long plate simply supported on all edges. The plate buckles when the compressive stress reaches sigma_cr.

• Compressive stress from a restrained temperature rise. Free thermal strain alpha * dT is blocked, so the restraint converts it into stress. Uniaxial restraint:

  sigma = E * alpha * dT
  

  Biaxial restraint (restraint in both in-plane directions):

  sigma = E * alpha * dT / (1 - nu)
  

• Critical temperature rise of a restrained plate: set the thermal stress equal to sigma_cr and solve for dT. Uniaxial:

  dT_cr = sigma_cr / (E * alpha)
  

  Biaxial:

  dT_cr = sigma_cr * (1 - nu) / (E * alpha)
  

• Euler column between rigid supports: the axial thermal load is P = alpha * E * A * dT and buckling occurs at P_cr = pi2 * E * I / L_eff2. With I = A * r**2 the area and modulus cancel:

  dT_cr = pi**2 * r**2 / (alpha * L_eff**2)
  

• Margin of a hot panel: margin = sigma_cr / sigma_thermal - 1. Positive means the panel is safe at that temperature rise; the sign flips when the rise exceeds the critical value.

• FAR 25 frames the airframe strength context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the material and geometry: elastic_modulus E, poisson nu, coefficient alpha, plate thickness t and width b, or column effective_length L_eff and radius_of_gyration r.
2. Compute the plate elastic buckling stress with plate_buckling_stress (k_coefficient defaults to 4.0).
3. Compute the compressive stress developed at the operating temperature rise: thermal_stress_uniaxial for restraint in one direction or thermal_stress_biaxial when both in-plane directions are blocked.
4. Find the temperature rise that buckles the panel with critical_temp_plate, passing restraint = "uniaxial" or "biaxial"; for a column between rigid supports use column_critical_temp.
5. Run the full check with thermal_buckling_assessment, which returns buckling_stress_Pa, thermal_stress_Pa, critical_temp_rise_K and the margin in one dict.
6. Read the margin: positive margin means the rise is below the critical value, negative means the panel buckles.
7. Confirm the deterministic checks with the contract test scripts/test_thermal_buckling.py.

Worked example

Aluminum skin panel: E = 72 GPa, nu = 0.33, alpha = 23e-6 /K, t = 1.6 mm, b = 150 mm, k = 4.0, uniaxial restraint.

• Plate buckling stress: sigma_cr = 30.24 MPa (30.244 MPa, inside the 25 to 40 MPa band of the hand estimate).
• Compressive stress at dT = 10 K: E * alpha * dT = 72e9 * 23e-6 * 10 = 1.656e7 Pa exactly (16.56 MPa).
• Critical temperature rise: uniaxial 18.26 K; biaxial 18.26 * (1 - 0.33) = 12.24 K, ratio exactly 0.67. Both inside the 15 to 25 K band for the uniaxial case.
• Margin at dT = 10 K: 30.24 / 16.56 - 1 = +0.83, panel safe. At dT = 30 K: 30.24 / 49.68 - 1 = -0.39, panel buckles.
• Euler column (steel, alpha = 12e-6 /K, r = 25 mm, L_eff = 2.0 m): dT_cr = pi2 * 0.0252 / (12e-6 * 2.0**2) = 128.5 K (110 to 150 K band).

Pitfalls

• Forgetting the restraint: thermal strain only becomes stress when the free expansion is blocked; an unrestrained panel heats without building load, so the uniaxial versus biaxial restraint choice is the physics, not a detail.
• Confusing the biaxial factor: biaxial restraint raises the stress by 1/(1 - nu) over the uniaxial case, so the biaxial critical rise is LOWER (12.24 K vs 18.26 K, ratio 0.67 exactly in the worked example) - blocking both directions makes the panel buckle sooner.
• Applying the k = 4.0 plate factor to other edge conditions: the buckling stress uses k = 4.0 for a long plate simply supported on all edges; other edge conditions carry other k coefficients and belong with the mechanical plate-buckling leaf.
• Checking the stress but not the margin sign: the thermal margin is sigma_cr / sigma_thermal - 1, positive below the critical rise and negative above (dT = 10 K gives +0.83, dT = 30 K gives -0.39 in the worked example); a positive stress is not the same as a safe panel.
• Reading the Euler column as plate-dependent: the column critical rise dT_cr = pi^2 r^2 / (alpha L_eff^2) is independent of the modulus because the area cancels - passing a stiffer material does not raise the column's critical temperature.
• Feeding non-physical inputs: non-positive modulus, thickness, width, length or k, poisson outside (-1, 0.5), alpha of zero in the critical-rise functions, a negative rise and invalid restraint strings all raise ValueError.

Verification

• Confirm plate_buckling_stress(72e9, 0.33, 1.6e-3, 0.150) returns 3.024e7 Pa, inside 25 to 40 MPa.
• Confirm thermal_stress_uniaxial(72e9, 23e-6, 10.0) returns exactly 1.656e7 Pa.
• Confirm critical_temp_plate returns 18.26 K uniaxial and that the biaxial value divided by the uniaxial value equals 0.67 within 1e-9 relative.
• Confirm column_critical_temp(200e9, 12e-6, 2.0, 0.025) returns 128.5 K and is independent of the modulus.
• Confirm the round trip: the thermal stress evaluated at the critical temperature rise equals the plate buckling stress.
• Confirm every non-positive modulus, thickness, width, length, k_coefficient, every poisson outside (-1, 0.5), alpha of zero in the critical rise functions, a negative temperature rise, an invalid restraint string, and a zero rise in the assessment raise ValueError.
• Run the contract test offline: python3 scripts/test_thermal_buckling.py (35 tests, deterministic).

Related leaves

• structures/thermal-structures/thermal-stress-analysis: constrained thermal stress of fully restrained members, bimetallic strips and their curvature; the companion leaf for members that do not buckle.
• structures/fem/plate-buckling: plate buckling under applied mechanical compression or shear with edge condition k coefficients.
• structures/materials/creep-rupture: material response limits for hot structure beyond elastic behavior.

Behavior contract (gate 3)

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

python3 scripts/test_thermal_buckling.py

The test covers the worked-example anchors (plate buckling stress 30.24 MPa in the 25 to 40 MPa band, exact 1.656e7 Pa thermal stress at dT = 10 K, critical temperature rise 18.26 K in the 15 to 25 K band, uniaxial-to-biaxial ratio 0.67, column 128.5 K in the 110 to 150 K band), the scaling identities of each closed-form relation, the round-trip identity between thermal stress at the critical rise and the buckling stress, positive and negative margin sign cases, determinism, and ValueError rejection of every non-physical input listed above.

Compliance

• Standards referenced, not reproduced: FAR 25 (airframe strength) is referenced by name only; the thermal buckling relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.