Thermal Loads & Constrained Expansion

What Are Thermal Loads?

A thermal load is a temperature change that causes a structure to develop stress. Temperature change causes thermal strain — materials expand when heated and contract when cooled. If the structure is free to expand and contract, the thermal strain produces deformation but no stress. If the structure is restrained — prevented from expanding or contracting by its supports, by adjacent structure or by its own geometry — the thermal strain cannot develop freely and stress is generated. Thermal stress also arises from non-uniform temperature distributions: if one part of a structure is hotter than another, the hot part tries to expand but is restrained by the cold part, and stress develops at the interface. Thermal loads are the loads that arise from temperature change in the presence of restraint or non-uniformity. They are not forces applied from outside — they are stresses generated internally by the conflict between thermal strain and geometric constraint.

TEMPERATURE CREATES THERMAL STRAIN. THERMAL STRESS DEVELOPS WHEN THAT STRAIN IS RESTRAINED OR NON-UNIFORM. A free body that is heated uniformly expands freely and develops no stress. A restrained body, or a body with a temperature gradient, develops stress.

Why Thermal Loads Matter

Thermal loads are significant in any structure that experiences temperature change: aerospace structures (aerodynamic heating, engine heat, solar heating, cryogenic propellants), engines and turbines (combustion temperatures, thermal gradients), electronic packages (power dissipation, thermal cycling), pipelines (hot fluid, ambient temperature changes), and composite structures (different coefficients of thermal expansion for different plies). In some cases, thermal stress is the governing load — a cryogenic tank during filling develops thermal stress from the temperature gradient through the wall; a turbine blade develops thermal stress from the temperature difference between the leading and trailing edges. Understanding when temperature becomes stress — and when it does not — is essential for correct load definition in thermal environments.

Thermal Strain and the CTE

The thermal strain is the deformation per unit length caused by a temperature change. It is the product of the coefficient of thermal expansion (CTE, or α) and the temperature change (ΔT). The CTE is a material property — it describes how much a material expands per degree of temperature change. Aluminium expands more than steel for the same temperature change (higher CTE); composites can have very low CTE in the fibre direction and higher CTE transverse to the fibres. The thermal strain is uniform in all directions for an isotropic material (same CTE in all directions) and directional for an anisotropic material (different CTE in different directions). The thermal strain is a strain — it is not a stress. It becomes a stress only when the strain is restrained.

Thermal strain:

  ε_thermal = α × ΔT

  where:
    ε_thermal = thermal strain (dimensionless, m/m)
    α = coefficient of thermal expansion (per K, or per °C)
    ΔT = temperature change (K, or °C)

  Note: ΔT in K = ΔT in °C (the interval is the same)

Thermal stress (fully restrained, uniform ΔT):

  σ_thermal = E × ε_thermal = E × α × ΔT

  where:
    σ_thermal = thermal stress (Pa)
    E = Young's modulus (Pa)
    α = CTE (per K)
    ΔT = temperature change (K)

  This is the stress if the structure is fully restrained —
  if it can expand freely, σ_thermal = 0.

  Partial restraint (stiffness ratio k):
    σ_thermal = E × α × ΔT × (k / (k + k_structure))

  where k = restraint stiffness, k_structure = structure stiffness
    k → ∞ (fully restrained):  σ → E × α × ΔT
    k = 0 (free):               σ → 0

Fully Restrained vs Free Expansion

The two extremes of thermal loading are fully restrained expansion and free expansion. In fully restrained expansion, the structure is prevented from expanding at all — the thermal strain is entirely converted to stress, and the stress is EαΔT. In free expansion, the structure is allowed to expand without any restraint — the thermal strain produces deformation but no stress. Real structures are typically somewhere between these extremes — partially restrained by their supports, by adjacent structure or by their own geometry. The degree of restraint determines the fraction of the thermal strain that becomes stress. A pipe anchored at both ends is fully restrained axially and develops the full EαΔT stress. A pipe anchored at one end is free to expand axially and develops no axial thermal stress. A pipe with a spring support develops partial thermal stress depending on the spring stiffness.

Thermal expansion — free vs restrained:

   Case 1: Free expansion (no stress)
   ───●─────────────────  (before heating)
   ───● ════════════════  (after heating, expanded)
      ↑
      free end — expands by α·ΔT·L, no stress

   Case 2: Fully restrained (full thermal stress)
   ●──┼─────────────────●  (before heating)
   ●──╫═════════════════●  (after heating, wants to
      ↑                  ↑   expand but cannot)
      fixed              fixed
      Stress = E × α × ΔT  (compressive if heated)

   Case 3: Partial restraint
   ●──┼─────────────────┼──  (before heating)
   ●──╫════════════════─╫──  (after heating)
      ↑                  ↑
      fixed          spring (k)

   Stress = E × α × ΔT × k / (k + k_bar)
   where k_bar = bar axial stiffness (EA/L)

   Key:  The restraint determines the stress.
         Temperature alone does NOT create stress.

Thermal Gradients and Non-Uniform Temperature

Even without external restraint, a non-uniform temperature distribution creates thermal stress. If one part of a structure is hotter than another, the hot part tries to expand more than the cold part. The two parts are connected — they must deform together — so the hot part is compressed by the cold part and the cold part is stretched by the hot part. This creates thermal stress even in a structure with no external restraints. A thermal gradient through a wall creates bending — the hot face expands more than the cold face, and the wall curves. If the wall is restrained against bending (by adjacent structure or by its own curvature), bending stress develops. A thermal gradient along a beam creates axial stress — the hot end is compressed, the cold end is stretched. The magnitude of the stress depends on the gradient, the CTE, the modulus and the geometry — and it can be as large as the stress from a fully restrained uniform temperature change.

Bimetallic and Composite Thermal Stress

When two materials with different CTEs are bonded together and heated, they try to expand by different amounts — but the bond prevents differential expansion. The result is thermal stress in both materials and curvature of the assembly. This is the principle of the bimetallic strip — used in thermostats and temperature sensors. In composite materials, the same effect occurs at the ply level: each ply has a different CTE in the fibre and transverse directions, and the plies are bonded together. A cross-ply laminate heated uniformly develops thermal stress in each ply because the 0° plies constrain the transverse expansion of the 90° plies and vice versa. This is residual thermal stress — it develops during manufacture (cure at elevated temperature, cool-down to room temperature) and is present in the laminate even before any external load is applied. The engineer must account for residual thermal stress when assessing the margin of a composite laminate — it can consume a significant fraction of the allowable before any mechanical load is applied.

Thermal Load Cases

Thermal load cases must be defined for the critical thermal environments the structure experiences. For an aircraft, these include: ground-hot (maximum temperature on the ground), cruise (aerodynamic heating at cruise Mach), descent (rapid cooling from cruise to sea level), cold-soak (prolonged exposure to high-altitude cold), and engine heat (local heating near the engine). For a cryogenic tank, the critical case is filling — the rapid temperature drop from ambient to cryogenic. For a satellite, the critical cases are sun-facing and eclipse — the temperature swings between full solar heating and deep-space cooling. Each thermal case produces a different temperature field, and each temperature field produces different thermal stress. The structural analysis must be run for each critical thermal case, often combined with mechanical loads (pressure, inertial) to assess the combined margin.

CONSTRAINT: Thermal load cases must cover the full thermal environment — not just the maximum temperature. A rapid temperature change can produce higher transient thermal stress than a steady high temperature, because the gradient through the thickness is larger during the transient.

Common Mistakes in Thermal Load Definition

COMMON MISTAKE: Applying a uniform temperature change to a structure with no restraint and expecting thermal stress. A uniform temperature change on a free body produces expansion but no stress. Thermal stress requires restraint, a gradient, or a material mismatch. If none of these is present, the thermal "load" is just a deformation.

Verification: Thermal Load Check

LOAD CHECK: For a thermal load case, verify that the temperature field is correctly mapped to the structural model. Check that the restraints in the model match the physical restraints — over-restraint produces artificial thermal stress. Confirm that the thermal stress in a free, uniform-temperature section is zero — if it is not, the model has artificial restraint.

Key Takeaways

• Temperature creates thermal strain: ε_thermal = αΔT. Stress develops only when that strain is restrained or non-uniform
• A free body with uniform temperature change expands freely and develops no stress — restraint or gradient is required for thermal stress
• Fully restrained thermal stress is σ = EαΔT; partial restraint produces partial stress depending on the restraint stiffness
• Thermal gradients create internal stress even without external restraint — the hot and cold parts constrain each other
• Composite laminates develop residual thermal stress during manufacture due to differential CTE between plies — this must be accounted for in margin assessment