Thermal Stress Calculator: Stress, Strain, Force and Expansion

What is thermal stress and when does it matter?

Thermal stress is the internal stress that develops in a solid body when a change in temperature causes it to expand or contract, but its movement is partially or fully prevented by external constraints or adjoining materials. When a member is free to deform it simply changes length, and no stress results. But when the ends are fixed, the thermal strain is converted into mechanical stress that can buckle rails, crack concrete, fracture semiconductor bonds, or fatigue pipe joints through repeated heating and cooling cycles. Practical examples include: steel railway rails that buckle in summer heat if expansion gaps are absent; bridge deck joints that close and crack in cold winters; turbine blade root stresses due to hot gas exposure; and solder joints on circuit boards stressed by the mismatch between chip and board expansion coefficients.

The formulas used in this calculator

Three quantities build on each other:

• Thermal strain: ε = α × ΔT, where α is the linear coefficient of thermal expansion (per degree) and ΔT is the temperature change. Strain is dimensionless, typically a few parts per million per degree for metals.
• Thermal stress (fully constrained): σ = E × ε = E × α × ΔT. Young's modulus E converts strain into stress. The sign convention: positive ΔT (heating) gives compressive stress in a constrained member; negative ΔT (cooling) gives tensile stress.
• Axial thermal force: F = σ × A, where A is the cross-sectional area. This is the reaction force at the fixed supports.
• Free thermal expansion: δ = α × L × ΔT, where L is the original length. This is how much the member would move if unconstrained.

How to use this calculator

Select metric (MPa, mm, degrees Celsius) or imperial (ksi, inches, degrees Fahrenheit) units. Then pick a preset material, or choose Custom and enter your own Young's modulus and CTE. Enter the temperature change: positive for heating, negative for cooling. Supply the member's original length and cross-sectional area (these are needed for thermal force and free expansion; leave them at defaults if you only need stress). Finally, set the constraint level from 0% to 100%. The results update as you type. The Show your work panel below the outputs traces every arithmetic step with your exact numbers, which is useful for checking hand calculations or learning the method for the first time.

Worked example: steel rail in summer heat

A structural steel rail (E = 200 GPa, α = 12 × 10⁻⁶ /°C) is 10 m long with a 7,500 mm² cross-section. Temperature rises 40 °C above the installation temperature, and the ends are fully fixed.

• Thermal strain: 12 × 10⁻⁶ × 40 = 480 microstrain
• Thermal stress: 200,000 MPa × 480 × 10⁻⁶ = 96 MPa (compressive)
• Thermal force: 96 MPa × 7,500 mm² = 720,000 N = 720 kN
• Free expansion: 12 × 10⁻⁶ × 10,000 mm × 40 = 4.8 mm

Limitations and engineering considerations

This calculator assumes a uniaxial, homogeneous, isotropic member that is linearly elastic across the full temperature range. Real-world complications include:

• Biaxial or triaxial stress: plates and shells restrained in two or three directions develop stress states that require the full thermoelastic constitutive equations, not just EαΔT.
• Temperature-dependent properties: Young's modulus and CTE both change with temperature, sometimes significantly. The values in the preset table are near room temperature. At elevated temperatures (above roughly 300 °C for steel) use temperature-corrected material data.
• Plastic deformation: if the computed stress exceeds the yield strength, the material deforms permanently and the elastic formula no longer applies. Repeated thermal cycling in this regime causes low-cycle fatigue.
• Composite members: when two materials with different CTEs are bonded together (bimetallic strips, reinforced concrete, PCB laminates), compatibility conditions and interface shear stresses require a more detailed analysis.