Formula and Method for Thermal Stress
When a material is free to expand or contract as its temperature changes, no internal stress develops — it simply gets longer or shorter. But when the same material is fully restrained (its ends fixed so it cannot move), the temperature change still "wants" to happen, and the restraint pushes back with an internal stress. For a fully restrained, linearly elastic member, that stress is σ = E × α × ΔT, where E is the material's Young's modulus (elastic modulus), α is its coefficient of linear thermal expansion, and ΔT is the temperature change (final temperature minus initial temperature).
How the calculation works
The calculator first finds the free (unconstrained) thermal strain the material would experience: ε = α × ΔT. If the member were allowed to expand or contract freely, this strain would produce a length change with zero stress. When the member is instead fully restrained — bonded to a rigid wall, welded between fixed supports, or embedded in a much stiffer surrounding material — that strain cannot physically occur, so Hooke's law (σ = E × ε) converts the "blocked" strain directly into stress: σ = E × α × ΔT. Multiplying that stress by the member's cross-sectional area gives the restraining force, F = σ × A, which is the force the supports must be able to resist.
Tension, compression, and sign convention
A temperature rise (ΔT > 0) in a fully restrained member produces compressive stress, because the material wants to expand but cannot. A temperature drop (ΔT < 0) produces tensile stress, because the material wants to contract but is held in place. This calculator reports the magnitude of the stress and states which type applies in the results panel. Always check the sign convention used in your own reference material, since some engineering texts define tensile stress as positive and compressive as negative, while others do the reverse.
Real-world applications and limits
Thermal stress calculations like this one govern the expansion gaps in bridges and railway track, the design of pipelines that run hot fluid through fixed supports, and the cracking risk in bonded assemblies, castings, and solder joints where two materials with different expansion coefficients are joined. This formula assumes full restraint and a linearly elastic, homogeneous material; real structures are often only partially restrained (which scales the stress down proportionally), and stresses that exceed the material's yield strength will cause permanent deformation rather than the purely elastic stress this formula predicts.