Thermal Stress Calculator

When a material is heated, its atoms vibrate more energetically and occupy more space, causing the material to expand. If this expansion is completely prevented by rigid supports, the material develops internal compressive stress (for heating) or tensile stress (for cooling). The fundamental relationships are:

$$\varepsilon_{th} = \alpha \, \Delta T$$

$$\sigma = E \, \varepsilon_{th} = E \, \alpha \, \Delta T$$

Key principles governing thermal stress:

• Sign convention: A positive ΔT (heating) with full constraint produces compressive stress (positive value here represents magnitude). Cooling produces tensile stress. The sign of the result indicates whether the stress is compressive or tensile.
• Partial constraint: In practice, most structures are neither perfectly free nor perfectly fixed. A constraint fraction of 0.5 means half the thermal strain is accommodated by expansion and half produces stress: $$\sigma_{partial} = E \, \alpha \, \Delta T \times f_c$$
• Material dependence: Steel (E ≈ 200 GPa, α ≈ 12 × 10⁻⁶/°C) develops much higher thermal stress than aluminum (E ≈ 70 GPa, α ≈ 23 × 10⁻⁶/°C) for the same ΔT, despite aluminum expanding nearly twice as much, because steel's stiffness dominates.
• Free expansion: If unconstrained, the material simply changes length by δ = α × ΔT × L per unit length. No stress develops in the absence of constraint.

Thermal stress analysis is essential in pressure vessel design, railroad track engineering, electronic packaging (solder joint fatigue from thermal cycling), dental restorations, and aerospace structures exposed to extreme temperature gradients. Combined thermal and mechanical loading requires superposition of stresses to check against material limits.

For anisotropic or composite materials, directional expansion coefficients must be used, and the stress state becomes multiaxial. In such cases, the biaxial thermal stress for a thin plate constrained in two directions is $$\sigma = \frac{E \, \alpha \, \Delta T}{1 - \nu}$$ where ν is Poisson's ratio.

The free thermal strain shows how much the material would deform if unconstrained — this is the strain that must be suppressed to generate stress. The thermal stress is the resulting internal stress when that strain is prevented. Compare this value to the material's yield strength: if σ exceeds σ_y, plastic deformation or cracking will occur. The free expansion per meter gives a practical measure of how much a one-meter bar would lengthen or shorten if free to move.