Tolerance Analysis
Will the parts still fit when every one of them is made a little too big or too small? List the dimensions that add up to a gap, clearance or overall length and compare the three standard methods side by side: worst case, root sum of squares (RSS, statistical) and a Monte Carlo simulation of 50 000 assemblies. Add the limits the result must meet for the expected share of rejects and the Cp/Cpk, and see which tolerance is worth tightening.
Worst Case or Statistical?
No part is made exactly to size, so every dimension comes with a tolerance. When parts are put together, their deviations add up. A tolerance stack-up follows the chain of dimensions that ends in the quantity you care about, such as the play of a shaft between its bearings, and asks how far that result can vary.
The worst case method adds all tolerances, as if every part were at its most unfavourable limit at the same time. The result is guaranteed, but with many parts that combination is extremely unlikely, so the tolerances end up tighter and more expensive than needed. The RSS method uses the fact that deviations of independent parts partly cancel: the spread of the sum grows with the square root of the sum of squares. With five equal parts, RSS gives √5 ≈ 2.2 times one tolerance instead of 5 times.
RSS assumes centred, independent processes with a normal spread. Real processes drift, so many companies use Bender’s factor 1.5 × RSS as a safety margin, or check with a Monte Carlo simulation that draws thousands of random assemblies. The variance share shows which part dominates: because the tolerances are squared, tightening the largest one helps by far the most.
Spread of the Result
Monte CarloThe Chain of Dimensions
nominal sizes, to scaleWhich Tolerance Matters Most?
| Part | +/− | Band | ±t | Worst-case share | Variance share |
|---|
Worked Techniques
Write each dimension as the middle of its band plus or minus half the band. Add the middles with their signs; the signs never matter for the tolerances, which always add up.
If each tolerance is ±3σ of a normal process, the result is again normal with ±3σ equal to this square root: 99.73% of the assemblies fall inside.
Cp compares the allowed width with the spread; Cpk also punishes an off-centre result. Cpk = 1 means the nearest limit is 3σ away (about 1350 ppm outside on that side), 1.33 means 4σ (32 ppm) and 1.67 means 5σ.
Sources & Verification
- B. R. Fischer, Mechanical Tolerance Stackup and Analysis, 2nd ed., CRC Press, 2011 — loop diagrams and sign conventions, worst-case and statistical (RSS) stack-ups.
- K. W. Chase and A. R. Parkinson, “A survey of research in the application of tolerance analysis to the design of mechanical assemblies”, Research in Engineering Design 3 (1991) 23–37 — worst case, RSS, inflated RSS and Monte Carlo compared.
- A. Bender, “Benderizing tolerances – a simple practical probability method of handling tolerances for limit stack-ups”, Graphic Science, December 1962 — the factor 1.5 on RSS.
- D. C. Montgomery, Introduction to Statistical Quality Control, 7th ed., Wiley, 2013 — process capability Cp and Cpk, fraction outside the limits.
- ISO 286-2:2010, Geometrical product specifications – ISO code system for tolerances on linear sizes – Part 2 — the limits of H7 and g6 at 10 mm used in the pin-and-hole example.