Thermal Loads & Imposed Strain

What Is It?

Thermal loads are structural forces and stresses created by temperature changes in a structure. Temperature itself is not a force — but when thermal expansion is restrained, the structure develops stress. A uniform temperature change in a free body produces expansion without stress. A uniform temperature change in a constrained body produces stress because the expansion is prevented. A temperature gradient (different temperatures at different points) produces stress because different parts expand by different amounts and the structure must remain continuous. A mismatch in coefficient of thermal expansion (CTE) between bonded materials produces stress at the interface. Thermal loads are unique — they create structural stress without a conventional applied force.

Why It Matters

Thermal loads are significant in many engineering applications. A gas turbine blade operates at high temperature and develops thermal stress from the temperature gradients. A satellite structure in orbit experiences extreme temperature changes as it moves between sun and shadow. A welded structure develops residual stress from the differential cooling of the weld and the surrounding material. A composite structure with different CTE plies develops micro-stress from temperature changes. In all these cases, the thermal stress may be comparable to or exceed the mechanical stress — and it must be included in the structural analysis. Ignoring thermal loads in a structure with significant temperature changes leads to under-prediction of stress and potential failure.

TEMPERATURE IS NOT A FORCE — BUT RESTRAINED THERMAL EXPANSION CAN CREATE FORCE AND STRESS. The thermal load is not the temperature itself but the stress created by the restraint of thermal expansion. A free body expands freely and has no thermal stress; a constrained body develops thermal stress. The restraint is what creates the load.

Thermal Strain

When a material is heated, it expands. The thermal strain — the strain from the temperature change — is the coefficient of thermal expansion times the temperature change. If the expansion is free (unconstrained), the thermal strain produces deformation but no stress. If the expansion is restrained, the thermal strain is converted to stress — the material tries to expand but cannot, and the restraint generates a compressive stress. The thermal stress is the elastic modulus times the difference between the free thermal strain and the actual strain (which is constrained).

Thermal strain:

  εth = α × ΔT

where:
  εth = thermal strain (dimensionless)
  α = coefficient of thermal expansion (1/K or ppm/K)
  ΔT = temperature change (K)

Free expansion (no stress):
  ε = εth = α × ΔT
  σ = 0

Fully restrained (no strain):
  ε = 0
  σ = −E × α × ΔT

where:
  E = elastic modulus (Pa)
  σ = thermal stress (Pa)

Uniform Temperature Change

A uniform temperature change — the whole structure at the same temperature — produces free expansion if the structure is unconstrained. A free bar heats up and expands lengthwise with no stress. If the same bar is fixed at both ends, the expansion is prevented and the bar develops compressive stress equal to E × α × ΔT. If the bar is part of a larger structure, the expansion is partially restrained by the surrounding structure — the stress depends on the stiffness of the restraint. A uniform temperature change in a statically determinate structure (no redundant constraints) produces expansion without stress. A uniform temperature change in a statically indeterminate structure (redundant constraints) produces stress because the redundant constraints prevent free expansion.

Thermal Gradient

A thermal gradient — different temperatures at different points in the structure — produces stress even in a free body. If one side of a beam is hotter than the other, the hot side expands more than the cold side, and the beam bends. If the bending is free (no external restraint), the beam curves but has no membrane stress — though there is a bending stress distribution through the thickness. If the bending is restrained (the beam is fixed), the thermal gradient produces both bending and membrane stress. Thermal gradients are common in heat exchangers, turbine blades (hot gas path vs. cooled interior), and structures with heat sources (electronics, engines). The thermal gradient stress depends on the gradient magnitude, the section depth and the restraint.

CTE Mismatch

When two materials with different coefficients of thermal expansion are bonded together, a temperature change produces stress at the interface. The material with the higher CTE expands more; the material with the lower CTE expands less. The bond prevents free differential expansion, creating stress in both materials. A common example is a bimetallic strip — two metals bonded together that bend when the temperature changes because one expands more than the other. In composites, the different plies may have different CTEs (particularly carbon fibre, which has near-zero or negative CTE in the fibre direction) and a temperature change produces micro-stress between the plies. In electronics, a silicon chip on a substrate with a different CTE develops stress at the solder joints under temperature cycling — a common failure mechanism.

Constrained Expansion and Imposed Displacement

A constrained expansion is one where the structure is prevented from expanding freely — the constraint generates stress. The constraint may be external (a fixed support) or internal (a stiffer part of the structure that prevents expansion of a softer part). The imposed displacement is a related concept — instead of applying a force, a displacement is imposed at a boundary (a settlement, a press fit, an interference). The imposed displacement generates stress in the structure because the structure is forced to accommodate the displacement. In FEA, the imposed displacement is applied as a boundary condition — the node is displaced by a specified amount and the structure develops stress to accommodate. The thermal load and the imposed displacement are analogous — both create stress through constraint rather than through an applied force.

Temperature Fields as Structural Inputs

The temperature field — the temperature at every point in the structure — is the input for thermal stress analysis. The temperature field may be uniform (a single temperature for the whole structure), a gradient (a linear variation through a dimension), or a full 3D field (different temperature at every point, typically from a thermal analysis). The temperature field is obtained from a thermal analysis (steady-state or transient) or from measurement. The structural model uses the temperature field to compute the thermal strain at each point and the resulting thermal stress. The mapping of the temperature field from the thermal mesh to the structural mesh is analogous to the pressure mapping in aerodynamics — the field must be transferred consistently and the total thermal strain must be preserved.

Common Mistakes

COMMON MISTAKE: Applying a uniform temperature change to a constrained structure and expecting no stress. A uniform temperature change in a constrained (statically indeterminate) structure produces stress. The stress depends on the restraint stiffness and the temperature change — it is not zero.

Key Takeaways

• Thermal strain is α × ΔT — free expansion produces no stress; restraint produces stress
• A uniform temperature change produces stress only in constrained (indeterminate) structures
• A thermal gradient produces bending stress even in a free body
• CTE mismatch between bonded materials creates stress at the interface under temperature change
• Temperature fields are structural inputs — mapped from thermal analysis to the structural model