What is the coefficient of thermal expansion (CTE)?

CTE (α) is the fractional change in length per degree of temperature change. Units: µm/(m·°C) or 10⁻⁶/°C. Example: steel α ≈ 12 µm/(m·°C) means a 1m bar grows 12 µm per 1°C rise.

How do I calculate thermal stress in a constrained member?

When expansion is fully restrained: σ = E × α × ΔT. For steel at ΔT = 100°C: σ = 200,000 × 12e-6 × 100 = 240 MPa — nearly yield stress!

What is volumetric expansion?

For isotropic materials, volumetric CTE ≈ 3 × linear CTE. Volume change: ΔV = V₀ × 3α × ΔT. Important for tanks, pipes, and enclosed fluid systems.

How do I size an expansion loop for piping?

Guided-cantilever formula: loop leg length L = √(3 × E × D × Δ / Sₐ), where E = elastic modulus (MPa), D = pipe OD (mm), Δ = expansion (mm), Sₐ = allowable stress range (MPa); L in mm. Rule of thumb for carbon steel (E ≈ 200 GPa, Sₐ ≈ 138 MPa): L ≈ 66 × √(D × Δ) in mm.

Does CTE vary with temperature?

Yes. CTE generally increases slightly with temperature. Values given are mean values over a range (typically 20–100°C). For cryogenic or high-temp work, use specific data for the actual temperature range.

Free Engineering Tool

Thermal Expansion Calculator

Calculate linear and volumetric thermal expansion for 20+ materials. Pipe expansion between anchors, expansion loop sizing, and thermal stress when constrained (σ = E·α·ΔT).

Linear & Volumetric20+ MaterialsThermal StressPipe Expansion

Material

CTE α (×10⁻⁶ /°C)

Original Length (mm)

Temperature Change ΔT (°C)

Original Volume (cm³ — optional, for volumetric)

Elastic Modulus E (GPa — for stress calc)

Quick presets

Expansion Results

Linear Expansion ΔL

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Expansion in mm

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Expansion in µm

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Expansion in mils

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New Length

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Strain ε

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Thermal Stress (if constrained)

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Thermal Force (per cm² area)

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Volumetric Expansion ΔV

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Expansion Loop (steel pipe, rule of thumb)

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Linear Expansion

Where α is the coefficient of thermal expansion (CTE) in 1/°C, L₀ is the original length, and ΔT is the temperature change.

Volumetric Expansion

For isotropic materials, the volumetric CTE is approximately 3× the linear CTE.

Thermal Stress (Constrained Member)

When expansion is completely prevented (e.g., pipe between two rigid anchors), the resulting stress can be very high. This is compressive for heating and tensile for cooling.

Expansion Loop Sizing (Rule of Thumb)

Where D is the pipe outer diameter (mm) and Δ is the expansion (mm); the result L is in mm. The factor 66 comes from the guided-cantilever formula L = √(3 · E · D · Δ / Sₐ) evaluated for carbon steel (E = 200 GPa, Sₐ = 138 MPa). For other materials or allowable stresses use the full formula (L scales as √(E/Sₐ)). This is a screening estimate, not a full code flexibility analysis.

Practical Example

Example — 6m Carbon Steel Pipe, ΔT = 80°C

Given: L₀ = 6000 mm, α = 12 × 10⁻⁶ /°C, ΔT = 80°C, E = 200 GPa

ΔL = 12e-6 × 6000 × 80 = 5.76 mm

Thermal stress if constrained: σ = 200,000 × 12e-6 × 80 = 192 MPa

Expansion loop (DN100/OD 114 mm): L ≈ 66 × √(114 × 5.76) ≈ 1,690 mm ≈ 1.7 m

Material CTE Database

Material	α (×10⁻⁶ /°C)	E (GPa)	Notes
Carbon steel	12.0	200	SA-516, P265GH
Low-alloy steel	12.5	200	16Mo3, SA-387
Cr-Mo steel (2.25Cr)	12.0	205	SA-335 P22
Stainless 304	17.3	193	1.4301, 18Cr-8Ni
Stainless 316	16.0	193	1.4401, 16Cr-10Ni-2Mo
Stainless 321	17.0	193	1.4541, Ti-stabilized
Duplex 2205	13.0	200	1.4462
Aluminium 6061	23.6	69	T6 temper
Copper	17.0	117	Pure, annealed
Brass (CuZn30)	20.0	110	Cartridge brass
Bronze (CuSn8)	18.0	110	Phosphor bronze
Titanium Gr.2	8.6	103	Commercially pure
Nickel 200	13.3	207	Pure nickel
Inconel 625	12.8	207	Ni-Cr-Mo
Inconel 718	13.0	211	Ni-Cr-Fe
Grey cast iron	10.5	110	GG25
Ductile cast iron	11.0	170	GGG40
Concrete	12.0	30	Varies with mix
Glass (soda-lime)	9.0	72	Window glass
PTFE (Teflon)	120.0	0.5	Very high CTE
HDPE	150.0	1.0	Polyethylene
PVC	80.0	3.0	Rigid PVC
Invar 36	1.2	145	Ultra-low CTE alloy

💡 Note: CTE values are mean values for 20–100°C. For high-temperature or cryogenic applications, use specific data from material standards.