Compute thermal stress (σ = α · E · ΔT), thermal strain, and restraining force for any material under constrained thermal expansion. Interactive stress-temperature graph included.
When a material is heated or cooled and prevented from expanding or contracting freely, internal stresses develop — these are thermal stresses. The governing equation for a fully constrained isotropic elastic body is: σ = α · E · ΔT, where σ is thermal stress (Pa or MPa), α is coefficient of linear thermal expansion (1/°C), E is Young's modulus (Pa), and ΔT = T_final − T_initial. This principle is critical for designing bridges, railway tracks, heat exchangers, electronic packaging, and high-temperature reactors.
σthermal = α · E · (T − T₀) and εthermal = α · ΔT
For bi-axial or tri-axial constraints, modified formulas apply. This calculator assumes 1D axial constraint (e.g., a bar fixed between rigid supports).
Modern railways use continuously welded rails that experience large temperature variations. For a steel rail (α=12×10⁻⁶/°C, E=200 GPa), a ΔT of 50°C generates σ = 12e-6 × 200e3 × 50 = 120 MPa. Without stress relief, this exceeds typical yield strength of rail steel (~350 MPa? safety factor considered). Engineers incorporate expansion joints or pre-stressing to manage thermal forces. Our calculator helps rail engineers estimate worst-case thermal loads.
Semiconductor packages often experience thermal cycling. Solder joints between silicon chip (α≈2.6) and PCB (α≈17) induce shear stresses. Using the thermal stress principle, design rules ensure fatigue life. This tool aids quick assessment of potential failure risks.
| Material | α (×10⁻⁶ /°C) | E (GPa) | Typical Yield Strength (MPa) | Max safe ΔT (approx) |
|---|---|---|---|---|
| Carbon Steel | 12.0 | 200 | 250 | 104 °C |
| Aluminum 6061 | 23.0 | 69 | 240 | 151 °C |
| Copper | 16.8 | 110 | 200 | 108 °C |
| Concrete | 10.0 | 30 | 20-40 (tension) | ~66 °C (cracking risk) |
| Stainless Steel 304 | 17.3 | 193 | 215 | 64 °C |
Data sources: CTE values from NIST SRM 738 and ASM International (Coefficient of Thermal Expansion of Solids), Young's moduli from ASM Metals Handbook Vol. 2 (Properties and Selection: Nonferrous Alloys). Additional verification against CES EduPack 2024.
Yield strength references: Typical values from ASME BPVC Sec. II-D and MMPDS-17. Actual values depend on temper, heat treatment, and product form.
The linear formula assumes homogeneous isotropic material, constant α and E over temperature range, and perfect elastic behavior. In reality, α and E vary with temperature, and very high ΔT may induce plastic deformation or creep. The calculator provides first-order engineering estimates; for design-critical systems, perform FEA and consult relevant codes (ASME BPVC Section VIII, ISO 21003). For piping systems, ASME B31.3 (Process Piping) requires detailed stress analysis including stress intensification factors and fatigue; this tool is suitable for preliminary screening but not for final certification. Additionally, for biaxial stress states (e.g., plates), σ = α·E·ΔT/(1-ν) (ν = Poisson's ratio). This tool focuses on uniaxial constraint, the most common simplified case.
Note on temperature-dependent properties: For large ΔT (>200°C), α and E typically change nonlinearly. Consult material handbooks (e.g., ASME BPVC Section II, Part D) for property curves and perform segmented calculations if high accuracy is required.
When a rod of length L is fully restrained, the thermal strain εth = αΔT is fully converted to mechanical strain εmech = −αΔT, leading to compressive stress σ = E·εmech = −EαΔT. The negative sign indicates compression upon heating, tension upon cooling. The magnitude is |σ| = αE|ΔT|. This is the fundamental basis for countless engineering designs. More formally, starting from Hooke's law: σ = E·(εtotal − εthermal). For a fully constrained body, εtotal = 0, hence σ = −E·α·ΔT. The absolute value is used for design checks.