Thermal Stress Analysis

Thermal Stress Analysis (structures/thermal-structures/thermal-stress-analysis)

Use when the task is the thermal stress and strain of a constrained aerospace structure: a member whose free expansion is blocked and therefore develops sigma = E * alpha * dT, a bonded bimetallic strip whose two layers bend under a temperature change, or a plate constrained against in-plane expansion that may buckle thermally. 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, dT in K, t and b in m, stresses in Pa, curvature in 1/m, force per unit width in N/m.

Domain quick reference

• Free thermal strain of an unrestrained member:

  eps = alpha * dT
  

  If the expansion is free, no stress develops. The stress appears only when the structure restrains the expansion.

• Fully constrained member (free expansion blocked in the load direction):

  sigma = E * alpha * dT
  

  Cooling (negative dT) reverses the sign and puts the member in tension instead of compression. The constraint can be geometric (a member welded or bolted between rigid supports) or kinematic (a thermal gradient in a redundant structure).

• Bimetallic strip of two bonded layers with thicknesses t1, t2, moduli E1, E2 and coefficients alpha1, alpha2 under a temperature change dT. The layers must share the interface strain, so an equal and opposite axial force per unit width P develops together with a common curvature kappa:

  kappa = (alpha2 - alpha1) * dT /
          ( (t1 + t2)/2 + 2*(E1*I1 + E2*I2)/(t1 + t2)
            * (1/(E1*t1) + 1/(E2*t2)) )
  P     = 2 * kappa * (E1*I1 + E2*I2) / (t1 + t2)
  

  with I_i = width * t_i^3 / 12 per unit width. The layer with the higher coefficient ends up in compression on the concave side. For equal thicknesses and equal moduli the result collapses to the classic forms kappa = 1.5 * (alpha2 - alpha1) * dT / (t1 + t2) and a layer stress of magnitude E * (alpha2 - alpha1) * dT / 8.

• Thermal buckling of a plate constrained against in-plane expansion (a long skin panel held between stiffeners): the temperature rise generates the compression E * alpha * dT, and buckling occurs when that stress reaches the plate critical stress, so

  dT_cr = k * pi^2 / (12 * (1 - nu^2) * alpha) * (t / b)^2
  

  with k the plate buckling coefficient (4.0 for a simply supported long plate, 6.97 clamped). The critical temperature rise scales with the square of the thickness-to-width ratio, exactly like the elastic buckling stress it is derived from.

Worked anchors (verified by running scripts/thermal_stress_analysis_logic.py): an aluminum member with E = 70 GPa, alpha = 23e-6 1/K and dT = 100 K has a free strain of 2300 microstrain and, fully constrained, a thermal stress of 161 MPa; against an allowable of 250 MPa the margin of safety is 0.553. A bimetallic strip of steel (alpha = 11e-6 1/K) and aluminum (alpha = 23e-6 1/K), each 1 mm thick with E = 70 GPa, under dT = 100 K bends to kappa = 0.9 1/m with a layer stress of 10.5 MPa. Doubling the aluminum modulus to 140 GPa changes the curvature to 0.8727 1/m and the layer stress to 15.27 MPa. A 2 mm aluminum skin on 150 mm stiffener pitch with E = 70 GPa, alpha = 23e-6 1/K and nu = 0.33 buckles at a critical temperature rise of 28.54 K.

Workflow

1. Decide what is being asked. A free member only needs the free strain; a member with blocked expansion needs the constrained stress; two bonded layers need the bimetallic balance; a constrained panel under heating needs the thermal buckling check.
2. For a constrained member, compute the free strain with free_thermal_strain(alpha, dT) and the stress with constrained_thermal_stress(E, alpha, dT). Run the complete margin check with thermal_stress_check(E, alpha, dT, allowable_stress), which returns the stress, the strain, the margin of safety and the acceptable verdict. Apply the required factor of safety from the certification basis before comparing against the allowable.
3. For a bonded pair, call bimetallic_strip(E1, E2, alpha1, alpha2, t1, t2, dT, width) and read off the curvature, the interface force per unit width and the two layer stresses. Use the returned force as the load input for a joint or bondline check.
4. For a constrained plate under a temperature rise, compute the critical temperature rise with thermal_buckling_critical_dT(E, alpha, nu, t, b, coefficient) or run the full check with thermal_buckling_check(E, alpha, nu, t, b, applied_dT, coefficient), which returns the critical temperature rise, the margin of safety and the stable verdict.
5. Verify the sign convention: positive dT with blocked expansion is compression; the layer with the higher coefficient sits on the concave side of a heated bimetallic strip.

Pitfalls

• Routing spacecraft thermal control here: radiator sizing, thermal balance and dissipation budgets of spacecraft subsystems belong to the space-systems thermal-design leaf; thermal-stress-analysis computes mechanical stress and strain from a temperature change in a constrained structure.
• Routing column thermal effects here: buckling-analysis handles 1D columns with Euler loads and slenderness; thermal-stress-analysis never uses a slenderness ratio and the thermal buckling check applies to a constrained flat plate only.
• Routing elastic plate buckling here: plate-buckling checks an applied mechanical edge load against the critical stress; thermal-stress-analysis derives the critical temperature rise from the same formula but the load comes from blocked thermal expansion.
• Forgetting the constraint: an unrestrained member carries no thermal stress no matter how large the temperature change; sigma = E * alpha * dT applies only when the expansion is blocked.
• Applying the biaxial correction blindly: a plate fully restrained in both in-plane directions develops E * alpha * dT / (1 - nu); the thermal buckling check here assumes the long-panel case restrained in one direction, which keeps the stress at E * alpha * dT. State the restraint model before choosing the formula.
• Mixing units: E in GPa with t and b in mm silently corrupts the stress and the critical temperature rise by factors of 1e9 or 1e6; keep everything SI (Pa, m).
• Using a negative coefficient: alpha must be positive for the thermal buckling check (the critical temperature rise divides by alpha); real materials with negative expansion coefficients need a dedicated treatment.
• Reading the bimetallic sign wrong: the layer with the higher coefficient is compressed and sits on the concave side; the two layer stresses are equal and opposite in magnitude when the thicknesses are equal.

Behavior contract (gate 3)

The thermal stress logic is exercised by the gate 3 contract test: scripts/test_thermal_stress_analysis.py against scripts/thermal_stress_analysis_logic.py (stdlib unittest, offline). It asserts the worked anchors above, the zero-stress case at dT = 0, the linear scalings with E, alpha and dT, the equal-layer bimetallic closed form and the unequal-modulus case, the (t/b)^2 scaling of the critical temperature rise, the margin and verdict outputs, and the ValueError cases for non-positive, non-finite or unknown inputs. Run:

python3 scripts/test_thermal_stress_analysis.py

Compliance

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