Thermal Expansion Calculator
Calculate linear and volumetric thermal expansion of materials.
Expansion Type
Material Presets (α)
What Is the Thermal Expansion Calculator?
Thermal expansion describes how materials change size when their temperature changes. Every solid, liquid, and gas expands when heated and contracts when cooled, this fundamental property drives engineering design across bridges, pipelines, rail tracks, and precision instruments.
Key Coefficients
- ›α (alpha), linear coefficient of thermal expansion (units: 1/°C or 1/K)
- ›Metals: steel ~12×10⁻⁶/°C, aluminum ~23×10⁻⁶/°C, copper ~17×10⁻⁶/°C
- ›Area expansion uses 2α (two-dimensional) and volumetric uses 3α (three-dimensional)
- ›Invar (nickel-iron alloy) has one of the lowest α values: ~1.2×10⁻⁶/°C
Constrained Expansion & Thermal Stress
- ›When expansion is blocked by a fixed constraint, thermal stress builds up
- ›σ = −E·α·ΔT (negative = compressive when heated)
- ›Young's modulus E for steel ≈ 200 GPa; aluminum ≈ 69 GPa
- ›Rail gaps and bridge expansion joints are engineered to absorb this stress
Coefficient values sourced from NIST Engineering Metrology Toolbox and ASM International Materials Data.
Formula
Linear Expansion
ΔL = α · L₀ · ΔT
Area Expansion
ΔA = 2α · A₀ · ΔT
Volumetric Expansion
ΔV = 3α · V₀ · ΔT
Final Length
L = L₀(1 + α·ΔT)
Thermal Stress
σ = −E · α · ΔT
Temperature Change
ΔT = T₂ − T₁
How to Use
- 1Select a material from the preset list or enter a custom α coefficient (×10⁻⁶/°C).
- 2Choose expansion type: Linear (1D), Area (2D), or Volumetric (3D).
- 3Enter the initial length/area/volume and the start (T₁) and end (T₂) temperatures.
- 4Toggle "Constrained Expansion" and enter Young's modulus to compute thermal stress.
- 5Click Calculate, results show ΔL/ΔA/ΔV, final dimension, and optional stress.
Example Calculation
Steel I-beam, L₀ = 10 m, heated from 0°C to 40°C, α = 12×10⁻⁶/°C:
ΔL = α × L₀ × ΔT
= 12×10⁻⁶ × 10 m × 40°C
= 0.0048 m (4.8 mm)
L_final = 10 + 0.0048 = 10.0048 m
Result
A 10-metre steel beam expands 4.8 mm over a 40°C temperature rise, exactly why bridge expansion joints must accommodate several centimetres across seasonal extremes. If fully constrained (E = 200 GPa): σ = −200×10⁹ × 12×10⁻⁶ × 40 = −96 MPa (compressive).
Understanding Thermal Expansion
Thermal Expansion Coefficients, Common Materials
| Material | α (×10⁻⁶/°C) | E (GPa) | Notes |
|---|---|---|---|
| Aluminum | 23.0 | 69 | Highly ductile; large expansion |
| Copper | 17.0 | 110 | Electrical wiring consideration |
| Steel (carbon) | 12.0 | 200 | Structural default |
| Stainless steel | 17.3 | 193 | Grade 304 |
| Cast iron | 10.8 | 170 | Lower than carbon steel |
| Brass | 19.0 | 100 | Alloy of Cu + Zn |
| Glass (soda-lime) | 8.5 | 70 | Fragile to thermal shock |
| Concrete | 12.0 | 30 | Matches steel, key for rebar |
| Invar | 1.2 | 148 | Precision instruments |
| Titanium | 8.6 | 116 | Aerospace structures |
| HDPE plastic | 150 | 0.8 | Much higher than metals |
| Fused silica | 0.55 | 73 | Ultra-low expansion optics |
Data sourced from NIST Engineering Metrology Toolbox and ASM International materials database.
Frequently Asked Questions
What is the coefficient of thermal expansion?
The linear CTE (α) measures expansion per unit length per °C rise. Common values:
- ›Steel (carbon): 12×10⁻⁶/°C, the structural engineering default
- ›Aluminum: 23×10⁻⁶/°C, nearly twice steel, important in composite structures
- ›Glass (soda-lime): 8.5×10⁻⁶/°C, lower, but brittle so thermal shock is critical
- ›Invar: 1.2×10⁻⁶/°C, ultra-low, used in precision instruments and clocks
What is the difference between linear, area, and volumetric expansion?
- ›Linear (1D): ΔL = α·L₀·ΔT, used for rods, beams, pipe lengths
- ›Area (2D): ΔA = 2α·A₀·ΔT, used for plates, membranes, surface coatings
- ›Volumetric (3D): ΔV = 3α·V₀·ΔT, used for blocks, tanks, fluid containers
- ›For isotropic materials the factor of 2 or 3 is exact; anisotropic materials require directional α values
How does thermal stress occur?
When a material tries to expand but is held fixed, the blocked expansion becomes internal stress:
- ›Formula: σ = −E·α·ΔT (negative sign means compressive when heated)
- ›Steel example: σ = −200 GPa × 12×10⁻⁶ × 40°C = −96 MPa
- ›−96 MPa approaches the yield strength of mild steel (≈250 MPa), real structural concern
- ›Bridge expansion joints, rail expansion gaps, and pipe loops prevent this buildup
Which material has the lowest thermal expansion?
- ›Invar (Fe-36Ni): ~1.2×10⁻⁶/°C, watch springs, surveying tapes, telescope mirrors
- ›Fused silica (SiO₂): ~0.55×10⁻⁶/°C, optical bench components, laser cavities
- ›Zerodur glass-ceramic: ~0×10⁻⁶/°C, telescope primary mirrors
- ›By comparison, HDPE plastic is ~150×10⁻⁶/°C, over 100× higher than steel
Why is thermal expansion important in civil engineering?
- ›A 100 m steel bridge expands ~14 mm from winter (−10°C) to summer (50°C), requires expansion joints
- ›Rail track "sun kinks": if expansion gaps are omitted, track buckles under summer heat
- ›Concrete and steel have nearly identical α (~12×10⁻⁶/°C), that's why rebar bonding works
- ›Thermal fatigue: repeated heating/cooling cycles cause cracking in constrained components over time
Does the formula work for negative temperature changes (cooling)?
Yes, the formula is fully bidirectional:
- ›Negative ΔT → negative ΔL = contraction (material shrinks)
- ›Example: aluminum rod cooled from 100°C to 20°C: ΔT = −80°C → contraction
- ›Cryogenic applications (liquid nitrogen at −196°C) see very large contractions
- ›Pipes carrying cold fluids contract; hangers and supports must accommodate this movement
How accurate is the linear approximation?
- ›Accurate to ~1% for metals over moderate ranges (0–300°C)
- ›α itself varies slightly with temperature, the linear model uses a constant average value
- ›For cryogenic use (below −100°C) or very high temperatures (above 500°C), tabulated α(T) curves are needed
- ›For precision engineering (optics, metrology), use the full integral ∫α(T)dT