Assembly tolerance realities.
Sum of tolerances
All dimensions at their worst limit simultaneously. 10 dimensions × ±0.1 = ±1.0 assembly tolerance. Conservative — accounts for manufacturing variation in all parts.
Root Sum Square
Assumes tolerances are normally distributed. Stack-up = √(Σ σ²). For same 10 × ±0.1: assembly ~ ±0.316. Much less conservative than worst case — 3× tighter.
Cost vs risk
Worst case: zero assembly failures, expensive parts. Statistical: rare assembly failures (often 1-2 per million), cheaper parts. Choose based on assembly criticality and cost tolerance.
Need many parts
Statistical analysis only valid at production volumes — need many parts to realize statistical distribution. For one-off prototypes, worst case is appropriate.
What adds up
Stack-up considers: part dimensions in the load path, mating feature tolerances, assembly fits and clearances, thermal expansion, form tolerances (flatness, parallelism).
Relax non-critical
Stack-up reveals which features are critical. Tighten critical dimensions, relax non-critical. Often enables 30-50% cost savings while maintaining assembly function.
Individual dimension tolerances combine to determine assembly tolerance. A stack of 10 dimensions each at ±0.1 mm does NOT give ±0.1 mm assembly tolerance.
Stack-up calculation.
Problem: Shaft through three bushings in housing. Need min clearance 0.1 mm to prevent binding, max clearance 0.5 mm to prevent excessive play. Five dimensions in stack: housing bore A (ØA ±tA), bushing 1 OD (Øb1 ±tb1), bushing 1 ID (ØB1 ±tB1), shaft diameter (ØS ±tS), assembly fit.
Worst case analysis: Max clearance = (housing bore at max) − (shaft at min). Min clearance = (housing bore at min) − (shaft at max). For housing bore Ø20 +0.021/0 (H7), bushing OD Ø20 -0.008/-0.021 (p6), bushing ID Ø12 +0.018/0 (H7), shaft Ø12 0/-0.011 (h6). Max clearance = 0.018 + 0.011 = 0.029 mm. Min clearance = 0 - (-0.011) = 0.011 mm. All within limits — assembly works at worst case.
Statistical (RSS) analysis: For same tolerance stack, σ_total = √(σA² + σb1² + σB1² + σS²). Assuming ±3σ = tolerance/2 (normal distribution), standard deviation for each dimension = tolerance/6. Computing: σ_total = √(0.0035² + 0.0022² + 0.003² + 0.0018²) = 0.0053 mm. Assembly ±3σ = ±0.016 mm — much tighter than worst case ±0.029 mm.
Practical implication: At production volumes, actual assembly variation ~±0.016 mm (statistical), not ±0.029 (worst case). If we designed based on worst case, we could loosen individual dimensions. If we redesigned each dimension +0.01 tolerance, new worst case ≈ 0.050 mm (still within 0.1-0.5 mm limit). New statistical = 0.022 mm. Production cost drops with looser tolerance, assembly still works reliably.
When to use which: Worst case: safety-critical, low-volume, when cost of assembly failure is high. Statistical: high-volume production, cost-sensitive, when some assembly failures are acceptable (and detectable). Aerospace typically uses worst case for safety features, statistical for non-safety features.
Analysis techniques.
Simple chains
Sum of 1D dimensions along axis. Spreadsheet or hand calculation. Adequate for most shaft/bore/length stacks.
Complex assemblies
Vector analysis for features at angles. CAD tools (SolidWorks TolAnalyst, Autodesk Inventor) automate. Required for complex mechanical assemblies.
Simulation
Generate random values per specified distributions, compute assembly for thousands of trials. Produces probability distribution of assembly variation. Best accuracy for non-normal distributions.
Feature-based
Stack-up considering GD&T tolerance zones (position, profile). More complex than ± tolerance stack-up but more realistic. Requires 3D analysis tool.
Dedicated tools
CETOL, VisVSA, 3DCS — dedicated tolerance analysis tools. Handle complex GD&T, statistical, Monte Carlo. Used for aerospace and automotive.
Reverse problem
Given assembly requirement, determine individual tolerances. Requires optimization — minimize cost while meeting assembly spec.
Stack-up rules of thumb.
When to do stack-up
Shortcuts & approximations
Common mistakes
Documentation
FAQ
For safety-critical assemblies (failure causes harm), use worst case. For cost-optimized production with some assembly failures acceptable, use statistical. For prototypes and one-off parts, worst case (no statistical reliability with n=1). For aerospace and medical, often use worst case for safety features and statistical for non-critical. Statistical requires process capability data — if you don't have it, stick with worst case.
Given assembly requirement, allocate individual tolerances. Equal allocation: divide assembly tolerance equally among dimensions. Weighted allocation: allocate more tolerance to dimensions cheaper to achieve loose (extruded, rough machining) and less to tight dimensions (reaming, grinding). Optimization tools (CETOL, spreadsheet solver) can minimize cost subject to assembly requirement. Usually tighter tolerance on few critical dimensions + looser on many non-critical gives best cost.
For assemblies operating outside room temperature, thermal expansion must be in stack-up. Each dimension changes with temperature: ΔL = L × α × ΔT, where α is coefficient of thermal expansion. Steel 12 ppm/°C, aluminum 23 ppm/°C, plastics 60-100 ppm/°C. For 100mm assembly cycling 80°C: steel grows 0.096 mm, aluminum 0.184 mm, plastic 0.5-0.8 mm. Can dominate other tolerances. Account for thermal drift in operating range.
Analytical (worst case, RSS): faster, easier, gives mathematical result. Limited to linear stacks and normal distributions. Monte Carlo: slower (needs simulation), handles non-normal distributions, complex geometry, non-linear relationships. For simple stacks, analytical is fine. For complex geometry (3D GD&T with form tolerances), Monte Carlo is more accurate and now accessible through CAD tools.
Typical margin: 20-30% of assembly tolerance between worst case stack-up and requirement. E.g., if requirement is 0.5 mm max clearance and worst case stack-up is 0.35 mm, margin is 0.15 mm (30% of requirement). Margin accounts for: unanticipated variation, future design changes, measurement error in verification, service conditions outside nominal. More margin = less risk but higher cost.
For complex mechanical assemblies with ISO 2768 or ± tolerances: 1D stack-up usually adequate. For assemblies using GD&T (position, profile, runout): 2D or 3D stack-up considering tolerance zones. GD&T stack-up is more realistic because it captures the actual allowed variation — position tolerance of 0.2 gives circular zone, ± 0.1 in X and Y gives square zone (27% more area). For precision assemblies, GD&T stack-up often shows more margin than equivalent ± analysis.
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