Tolerance Stack-Up Calculator — RSS Method, Worst-Case & Statistical Analysis
Calculate assembly tolerance stack-up using RSS (Root Sum Square) and worst-case methods. Predict assembly gap, interference, and statistical variation.
Quick Answer
For 5 components with ±0.1mm tolerance each in series: Worst-Case Stack = ±0.5mm, RSS Stack = ±0.224mm. The RSS method is 2.2× tighter and far more realistic for production — it assumes tolerances follow normal distribution and are independent.
How Tolerance Stack-Up Analysis Works
Tolerance stack-up predicts whether your parts will assemble and function. Two methods compete:
1. Worst-Case (WC) Method
ΔY_wc = Σ ΔXᵢ. Simply add all tolerances linearly. This guarantees 100% assembly — every part combo works. But it’s extremely conservative. For 10 parts at ±0.1mm each, worst-case requires ±1.0mm gap. RSS says ±0.316mm. Which one do you design to?
2. Root Sum Square (RSS) Method
ΔY_rss = √(Σ ΔXᵢ²). This is the statistical stack — it assumes tolerances are normally distributed and independent. For production volumes over 100 parts, this predicts reality. The tradeoff: ~0.27% of assemblies may fall outside the RSS band (3σ).
3. When Each Method Applies
WC: safety-critical (brake pedal clearance), low volume (<10 assemblies), or when a single out-of-spec assembly is catastrophic. RSS: high volume production, interchangeable parts, non-safety-critical assemblies. Most automotive and consumer products use RSS.
Applications
- Ensuring bearing fits and preload in assembled shafts
- Verifying gear mesh backlash across tolerance ranges
- Checking PCB-to-enclosure clearance in electronics
- Validating snap-fit and press-fit assembly feasibility
Common Mistakes
- Applying RSS when tolerances aren’t independent — If two dimensions come from the same machining setup, their errors correlate. RSS assumes independence. For correlated tolerances, use WC on those pairs, RSS on the rest.
- Using ±3σ when processes aren’t capable — RSS assumes Cp≥1.0 (process spread ≤ tolerance band). If your supplier runs at Cp=0.7 (2σ within tolerance), RSS underestimates stack by ~30%. Verify process capability before choosing method.
- Forgetting geometric tolerances — Flatness, parallelism, and runout stack like dimensions but with different statistical distributions. A ±0.05mm flatness tolerance acts differently than a ±0.05mm length dimension — more like a triangle distribution.
- Not including assembly variation — Press fits change dimensions. A bearing pressed into a housing compresses the outer ring 0.01-0.03mm. Add this to your stack or you’ll be chasing ghosts during assembly.
- Designing only for mean conditions — The assembly drawing works when everything is nominal. But production parts vary. Always run the stack at extreme combinations — that’s where problems hide.
Frequently Asked Questions
What is the difference between RSS and Monte Carlo simulation?
RSS gives a single 3σ number assuming normal distributions. Monte Carlo runs thousands of random combinations and gives a distribution — it can handle non-normal distributions and correlations. RSS is a 30-second calculation; Monte Carlo needs software. For most mechanical assemblies, RSS is sufficient.
How do I handle unilateral tolerances?
Convert to bilateral equivalent: a shaft Ø50 +0.05/0 becomes Ø50.025 ±0.025. The mean shifts. For stack-up, always work with symmetric bilateral tolerances centered on the mean dimension.
What Cpk should I assume for my supplier?
If you don’t know: assume Cpk=1.0 (industry average for general machining). Better suppliers: Cpk≥1.33. Precision grinding shops: Cpk≥1.67. For critical dimensions, require Cpk≥1.33 in the purchase spec and validate with incoming inspection.
When should I use worst-case instead of RSS?
Use WC when: (1) Batch size <10 assemblies (statistics don't apply), (2) One failure = catastrophic (aircraft control linkage, medical implant), (3) Tolerances are correlated (same setup), (4) Regulatory requirement (FDA often requires WC), (5) You can afford the extra cost of tighter tolerances.
How does GD&T affect stack-up analysis?
GD&T adds geometric variation on top of dimensional. Position tolerance (±0.1) stacks with size tolerance (±0.05). The combined effect is larger than either alone. For precise stacks, use 3D tolerance analysis software, not 1D RSS.
What if my assembly has a mix of tight and loose tolerances?
The RSS formula handles this naturally — tight tolerances contribute little to the total. A 0.01mm tolerance contributes only 1% of what a 0.1mm tolerance contributes to the RSS sum (0.01² = 0.0001 vs 0.1² = 0.01). Focus tolerance improvement on the largest contributors.
How do temperature changes affect the stack?
Different materials expand differently. An aluminum housing (CTE 23 ppm/°C) against a steel shaft (CTE 12 ppm/°C) changes clearance by (23-12)×L×ΔT. A 100mm length at ΔT=50°C shifts 0.055mm — swamping a typical ±0.02 tolerance. Always include thermal effects in the stack.
Can I relax tolerances to reduce cost?
Yes — that’s the power of RSS. If your WC stack says you need ±0.01mm on every part, but RSS says you can tolerate ±0.03mm, you just saved 50% on manufacturing cost. But validate with capability data first. Changing tolerances without process data is gambling.