Thermal Expansion in Tolerance Stack-Ups

Parts grow and shrink with temperature. When materials with different coefficients of thermal expansion (CTE) are assembled together, their relative sizes change with temperature—affecting fits, clearances, and preloads. This guide shows how to include thermal effects as signed contributors in your tolerance stack-up.

Model Thermal Effects

TolReport lets you add thermal expansion as dimensions in your stack. Calculate how temperature changes affect clearances, endplay, and interference fits.

The Thermal Expansion Formula

Linear thermal expansion follows a simple relationship:

ΔL = α · L · ΔT

Where:

ΔL — Change in length (mm or inches)
α — Coefficient of thermal expansion (CTE), typically in ppm/°C or μm/m·°C
L — Original length at reference temperature
ΔT — Temperature change from reference (T_operating − T_reference)

Common CTE Values

Material	CTE (ppm/°C)	Notes
Steel (carbon/alloy)	11–13	Shafts, pins, fasteners
Stainless steel (304/316)	16–17	Higher than carbon steel
Aluminum (6061)	23–24	Housings, brackets
Brass	18–21	Bushings, fittings
Invar	1–2	Low-expansion alloy
PEEK / POM	40–80	Engineering plastics

Units Matter

CTE in ppm/°C means "parts per million per degree Celsius." For a 100 mm part with α = 23 ppm/°C and ΔT = 50°C: ΔL = 23 × 10⁻⁶ × 100 × 50 = 0.115 mm. Always convert CTE to consistent units before calculating.

CTE Mismatch: The Real Problem

Uniform thermal expansion rarely causes problems—everything grows together. The problem is CTE mismatch between different materials in the same assembly.

Consider a steel shaft in an aluminum housing:

Aluminum (α ≈ 23 ppm/°C) expands about 2× faster than steel (α ≈ 12 ppm/°C)
As temperature rises, the housing bore grows faster than the shaft
Clearance increases with temperature
A press fit at room temperature may become a clearance fit when hot

The opposite happens when cooling: clearance decreases, and a running fit may become an interference fit.

Temperature Range vs. Assembly Temperature

Every thermal stack-up needs clear temperature definitions:

Reference temperature (T_ref) — Usually 20°C (68°F), the standard for dimensional inspection
Assembly temperature (T_assy) — Temperature when parts are assembled
Operating range (T_min to T_max) — Temperatures the assembly sees in service

You typically analyze two cases:

Hot case: ΔT = T_max − T_ref
Cold case: ΔT = T_min − T_ref (negative ΔT)

Assembly at Non-Standard Temperature

If parts are assembled hot (e.g., shrink fit) or cold (e.g., cryo assembly), your reference may be the assembly temperature, not 20°C. The dimensions on your drawing are at inspection temperature, but the fit you achieve depends on assembly conditions.

Adding Thermal Terms to the Stack

Thermal expansion enters the stack as additional dimension contributors. Each part that changes length with temperature adds a ΔL term.

Sign Convention

Use the same sign logic as dimensional contributors:

+ if expansion increases the gap/clearance
− if expansion decreases the gap/clearance

For a shaft-in-housing clearance stack:

Housing bore expansion → + (bore gets larger, clearance increases)
Shaft diameter expansion → − (shaft gets larger, clearance decreases)

Thermal "Tolerance"

Thermal expansion isn't a manufacturing tolerance—it's a deterministic shift based on temperature. You can model it two ways:

Fixed shift: Calculate ΔL for a specific ΔT and add it to the nominal stack
As a range: Calculate ΔL for T_min and T_max, treat the range as a "tolerance" around the nominal

The second approach lets you combine thermal effects with manufacturing tolerances in one WC/RSS analysis.

Bearing Endplay Interaction

Thermal effects are especially critical in bearing endplay stacks. A shaft/housing assembly typically has:

Steel shaft and steel bearings (similar CTE)
Aluminum or cast iron housing (different CTE)
Spacers of various materials

As temperature changes:

Scenario	Housing CTE vs Shaft	Endplay Effect
Heating	Aluminum > Steel	Endplay increases
Cooling	Aluminum > Steel	Endplay decreases (may become preload)
Heating	Cast iron ≈ Steel	Minimal change

This is why aluminum gearbox housings often require more endplay at assembly—to ensure they don't develop excessive preload when cold.

Worked Example: Steel Shaft in Aluminum Housing

A gearbox has a 200 mm long steel shaft supported by two bearings in an aluminum housing. Operating temperature ranges from −20°C to +80°C. Assembly is at 20°C.

Given Data

Parameter	Value
Housing bore spacing	200.00 ± 0.05 mm
Shaft bearing seat spacing	199.85 ± 0.03 mm
Bearing widths (total)	30.00 ± 0.04 mm
Housing material	Aluminum, α = 23 ppm/°C
Shaft material	Steel, α = 12 ppm/°C
Reference temperature	20°C

Dimensional Stack at 20°C (No Thermal)

Endplay = Housing spacing − Shaft spacing − Bearing widths

Nominal: 200.00 − 199.85 − 30.00 = −29.85 mm

Wait—that's not right. Let's reconsider the geometry...

Actually, the stack should be: Housing internal length minus (shaft shoulder-to-shoulder length + bearing widths). Let's use clearer numbers:

Dimension	Nominal (mm)	Tolerance	Sign
Housing bore length	65.00	±0.05	+
Bearing A width	15.00	±0.02	−
Spacer (steel)	34.85	±0.03	−
Bearing B width	15.00	±0.02	−

Endplay @ 20°C = 65.00 − 15.00 − 34.85 − 15.00 = 0.15 mm

WC tolerance = ±(0.05 + 0.02 + 0.03 + 0.02) = ±0.12 mm

Thermal Contributions

Now add thermal expansion. For the hot case (80°C, ΔT = +60°C):

Housing (aluminum):

ΔL = 23 × 10⁻⁶ × 65 × 60 = +0.090 mm (bore grows, +)

Steel spacer:

ΔL = 12 × 10⁻⁶ × 34.85 × 60 = +0.025 mm (spacer grows, −)

Bearings (steel):

ΔL = 12 × 10⁻⁶ × 30 × 60 = +0.022 mm (bearings grow, −)

Net thermal effect on endplay at 80°C:

ΔEndplay = +0.090 − 0.025 − 0.022 = +0.043 mm

For the cold case (−20°C, ΔT = −40°C):

Housing: −0.060 mm

Spacer: −0.017 mm

Bearings: −0.014 mm

ΔEndplay = −0.060 + 0.017 + 0.014 = −0.029 mm

Combined Analysis

Condition	Nominal Endplay	WC Range
20°C (assembly)	0.15 mm	0.03 – 0.27 mm
80°C (hot)	0.19 mm	0.07 – 0.31 mm
−20°C (cold)	0.12 mm	0.00 – 0.24 mm

Cold Case Risk

At −20°C worst case, endplay drops to 0.00 mm—borderline preload. If spec requires minimum 0.05 mm endplay at all temperatures, this design fails. Options: increase nominal spacer length, or use a housing material closer to steel's CTE.

Common Pitfalls

Ignoring thermal entirely — Many stack-ups assume room temperature. If your product sees a wide temperature range, this is a critical oversight.
Using average CTE — CTE varies with temperature. For extreme ranges, use temperature-specific values or integrate.
Forgetting the reference — Drawing dimensions are at inspection temperature (usually 20°C). Thermal growth is relative to that baseline.
Mixing materials carelessly — A steel bolt clamping aluminum parts will see increasing tension when heated. The bolt stretches less than the parts compress axially.
Assuming uniform temperature — In real assemblies, parts reach operating temperature at different rates. Transient thermal gradients can be worse than steady-state.

Key Takeaways

ΔL = α · L · ΔT — Simple formula, but apply it to every contributor
CTE mismatch drives problems — Uniform expansion is usually fine; mixed materials are not
Analyze hot and cold cases — Each extreme may be the limiting condition
Sign convention still applies — Expansion that increases gap is +, decreases is −
Thermal shifts are deterministic — Not a tolerance, but can be modeled as a range for combined analysis

Add Thermal Effects to Your Stack

TolReport lets you model thermal expansion as additional dimensions. See how temperature changes affect your clearances, endplay, and fits across the operating range.