Thermal Expansion Formula
Thermal expansion formula ΔL = αL₀ΔT predicts how materials expand with heat.
Covers linear and volumetric expansion for steel, aluminum, concrete, and glass.
The Formula
This formula calculates how much a material's length changes when its temperature changes.
Most materials expand when heated and contract when cooled.
The amount of expansion depends on the material's thermal expansion coefficient.
Variables
| Symbol | Meaning |
|---|---|
| ΔL | Change in length (metres, m) |
| α | Coefficient of linear thermal expansion (per °C or per K) |
| L₀ | Original length (metres, m) |
| ΔT | Change in temperature (°C or K) |
Common Expansion Coefficients
| Material | α (× 10⁻⁶ per °C) |
|---|---|
| Steel | 12 |
| Aluminium | 23 |
| Copper | 17 |
| Concrete | 12 |
| Glass | 8.5 |
Example 1
A 10 m steel bridge beam is heated from 15°C to 45°C. How much does it expand? (α = 12 × 10⁻⁶ /°C)
ΔT = 45 - 15 = 30°C
ΔL = α × L₀ × ΔT
ΔL = 12 × 10⁻⁶ × 10 × 30
ΔL = 0.0036 m = 3.6 mm
Example 2
A 2 m aluminium rod is cooled from 80°C to 20°C. Find the change in length. (α = 23 × 10⁻⁶ /°C)
ΔT = 20 - 80 = -60°C
ΔL = 23 × 10⁻⁶ × 2 × (-60)
ΔL = -0.00276 m = -2.76 mm (the rod contracts)
When to Use It
Use the thermal expansion formula when you need to:
- Design expansion joints in bridges, railways, and pipelines
- Account for thermal stress in constrained components
- Calculate clearance requirements for parts that operate at varying temperatures
- Predict dimensional changes in precision-machined components
For area expansion, multiply by 2α.
For volume expansion, multiply by 3α.
Key Notes
- Linear expansion: ΔL = α × L₀ × ΔT: α is the coefficient of linear thermal expansion (material constant, units: 1/°C or 1/K). Steel: α ≈ 12×10⁻⁶/°C; aluminum: ≈ 23×10⁻⁶/°C; invar (low-expansion alloy): ≈ 1×10⁻⁶/°C.
- Volumetric expansion: ΔV ≈ 3α × V₀ × ΔT: The volumetric expansion coefficient β ≈ 3α for isotropic solids. Liquids use β directly (water: β ≈ 207×10⁻⁶/°C at 20°C, but anomalously contracts between 0°C and 4°C — why ice floats).
- Expansion joints prevent structural failure: A 100 m steel bridge heated by 30°C expands ΔL = 12×10⁻⁶ × 100 × 30 ≈ 36 mm. Without expansion joints, this generates enormous compressive stress that can buckle the structure. Railroad tracks and concrete pavements have the same requirement.
- Bimetallic strips — differential expansion in action: Two metals with different α values bonded together curve when heated. The side with higher α is longer, so the strip bends toward the lower-α side. Used in thermostats, circuit breakers, and temperature gauges.
- Applications: Thermal expansion calculations govern bridge and railway joint design, pipeline expansion loops, precision machined parts (gauge blocks must be used at a reference temperature), glass-ceramic cooktops (ultra-low α prevents cracking), and telescope mirror design.