Thermal Stress Analysis

What Is It?

Thermal stress analysis is the prediction of stress in a structure caused by temperature changes. Temperature changes produce thermal strain (expansion or contraction). When this strain is restrained — by boundary conditions, by connections to other parts, by gradients within the structure or by material mismatch — it cannot be freely accommodated and stress results. Thermal stress analysis takes the temperature field from the thermal analysis and computes the resulting stress field, accounting for the structural restraints, the material properties and the geometry. It is the link between the thermal model and the structural assessment.

Why It Matters

Thermal stress can be the governing stress in many engineering structures — engines, aerospace structures, electronics, piping, pressure vessels. A structure that is fully adequate for mechanical loads may fail under thermal stress because the temperature field creates a stress state that the mechanical loads alone do not. Understanding how temperature becomes stress — through restraint, through gradients, through material mismatch — is essential for credible structural assessment. Thermal stress is not just about the temperature level; it is about what the temperature does to the structure.

High temperature does not automatically mean high thermal stress — restraint and gradients determine the structural response. A uniform temperature in a free body produces no stress. A small gradient in a restrained body can produce significant stress. The analysis must capture the restraint and the gradient, not just the temperature.

Restrained Expansion

The simplest thermal stress case is a fully restrained bar. The bar is heated (ΔT > 0) and wants to expand, but it is held fixed at both ends. The thermal strain that would occur (α·ΔT) is prevented — the bar cannot get longer. The prevented strain becomes stress. For a one-dimensional elastic bar with perfect restraint, the thermal stress is the product of the modulus, the CTE and the temperature change.

Fully restrained one-dimensional elastic thermal stress:

σ  ≈  E · α · ΔT

where:
σ     =  thermal stress (compressive for heating, tensile for cooling)
E     =  Young's modulus
α     =  coefficient of thermal expansion
ΔT    =  temperature change from reference

Assumptions: full restraint, elastic, constant properties, 1D

This is the MAXIMUM thermal stress for 1D — partial restraint gives less

Real structures are more complex: multi-dimensional, partial restraint,
gradients, multi-material, possible plasticity

Why Real Structures Are More Complex

The simple EαΔT formula is the maximum thermal stress for a one-dimensional, fully restrained, elastic case. Real structures are more complex in several ways. The restraint is rarely perfect — the structure can deform, reducing the stress. The stress is multi-dimensional — the structure expands in all directions, and the restraint in one direction affects the stress in others. The temperature is non-uniform — gradients create differential expansion and bending. The structure may be multi-material — different CTEs create differential stress. The stress may exceed yield — plasticity redistributes the stress. The simple formula is a useful upper bound and sanity check, but the real analysis requires FEA.

Partial Restraint

Most structures are partially restrained — they can deform but not freely. A panel bolted at its edges can expand toward the centre but not at the edges. A tube connected to a structure can expand axially but is restrained radially. The partial restraint allows some of the thermal strain to be accommodated by deformation, reducing the stress compared to full restraint. The actual stress depends on the stiffness of the restraint relative to the stiffness of the expanding part — a stiff restraint (thick wall, rigid support) gives stress close to full restraint; a flexible restraint (thin wall, compliant support) gives lower stress.

Multi-Material Assemblies

In a multi-material assembly, the differential expansion between materials with different CTEs creates stress. A steel bolt in an aluminium structure: the aluminium expands more than the steel when heated, creating compression in the aluminium and tension in the bolt. A composite panel bonded to a metal frame: the metal expands more than the composite (in the fibre direction), creating shear stress in the bond line. The stress depends on the CTE difference, the temperature change, the stiffness of each material and the geometry of the interface. Multi-material thermal stress is one of the most common and most important thermal-structural problems.

Thermal Gradients and Stress

A temperature gradient — non-uniform temperature — creates differential expansion within a single material. The hot region expands more than the cold region. If the structure is free, it bends — the differential expansion produces curvature, but no membrane stress. If the structure is restrained against bending, the curvature is prevented and bending stress results. If the structure is restrained against membrane expansion, membrane stress results in addition to the bending stress. The stress from a gradient depends on the gradient steepness, the restraint conditions and the structure stiffness.

Thermal Preload and Bolt Preload

Bolt preload changes with temperature. A bolted joint is preloaded at assembly temperature — the bolt is tensioned, clamping the joint members. When the temperature changes, the bolt and the joint members expand by different amounts (different CTEs or different temperatures), changing the preload. If the bolt has a higher CTE than the joint members and the temperature increases, the bolt expands more than the joint — the preload increases. If the joint members expand more, the preload decreases. The preload change affects the joint behaviour — clamping force, sealing, fatigue, friction. The thermal analysis must provide the temperature of the bolt and the joint members to compute the preload change.

COMMON MISTAKE: Computing thermal stress as E·α·ΔT without checking whether the structure is actually fully restrained. Real structures are partially restrained — the actual stress may be significantly lower than the full-restraint value. Use FEA to capture the actual restraint.

Plasticity and Creep at Elevated Temperature

At elevated temperature, the thermal stress may exceed the yield strength — and the yield strength itself is lower at high temperature. When thermal stress exceeds yield, the structure deforms plastically — the stress is redistributed, residual stress remains after cooling, and repeated thermal cycling can cause thermal fatigue (low-cycle fatigue). At very high temperature, creep — time-dependent deformation under sustained stress — becomes significant. Thermal creep can relax the thermal stress over time (stress relaxation) but also produces accumulated deformation. For structures at elevated temperature, the thermal stress analysis must include plasticity and possibly creep.

Mapping Thermal Results into Structural Model

The thermal stress analysis requires the temperature field from the thermal analysis to be mapped onto the structural model. The temperature at each point in the structure becomes a thermal load in the stress analysis. The mapping may be direct (same mesh for thermal and structural) or it may require interpolation (different meshes). The reference temperature must be consistent — the structural model must know the stress-free temperature to compute the thermal strain. The material properties must be appropriate for the temperature — temperature-dependent properties may be needed. The mapping must be verified — the total thermal strain in the structural model should be consistent with the temperature field from the thermal model.

Key Takeaways

• Thermal stress arises from restrained thermal expansion — full restraint gives σ ≈ EαΔT
• Real structures have partial restraint, gradients and multi-material mismatch — FEA is needed
• High temperature does not automatically mean high stress — restraint and gradients determine the response
• Bolt preload changes with temperature — differential CTE or temperature changes the clamping force
• At elevated temperature, plasticity and creep may need to be included in the thermal stress analysis