Tolerance Stack-Up Analysis: Worst Case vs RSS, Worked Through a Three-Part Assembly

Your assembly has a 0.40 mm window for end clearance. Three parts make up the stack, and each drawing looks reasonable on its own. Then a prototype binds in one unit out of five, and the argument starts: the machinist says the parts are in tolerance, the designer says the stack should work, and both are right.

That gap, between parts that are individually conforming and an assembly that sometimes does not fit, is what a tolerance stack-up exists to close. It converts a set of separate tolerances into one predicted assembly outcome, and it shows which of them is actually driving the result.

Two questions, two methods

Worst case assumes every dimension lands at its limit in the direction that makes the outcome worst. It is arithmetic: add the tolerances. If the result fits inside the specification, the assembly cannot fail, whatever the process does.

RSS, root-sum-square, assumes each dimension varies independently around nominal, roughly normally, and that three dimensions landing at their extremes at once is vanishingly unlikely. It combines the tolerances as the square root of the sum of squares. The result is a statistical spread with a probability attached, not a hard limit.

On a four-part stack with ±0.10 mm on each part, worst case gives 0.80 mm of variation and RSS gives about 0.20 mm at three sigma. That is the difference between milling a bore and grinding it.

A worked three-part stack

Take a small gearbox: a housing with a bored pocket, a bearing seated in the pocket, and a spacer that sets the axial position of a shaft. The requirement is end clearance between 0.10 mm and 0.50 mm, so nominal sits at 0.30 mm inside a 0.40 mm window.

The stack is linear: gap = housing pocket depth − bearing width − spacer length.

Sensitivity is the multiplier each dimension carries in the stack equation. Every part here contributes at unit sensitivity, which is typical of an axial stack. In a lever arrangement a small angular error can carry a sensitivity of three or four at the far end, and the table would show it.

Running worst case

Put A at its high limit and B and C at their low limits for the maximum gap, then reverse everything for the minimum.

Maximum: 22.40 − 11.97 − 9.95 = 0.48 mm Minimum: 22.20 − 12.03 − 10.05 = 0.12 mm

The full range is 0.36 mm against a 0.40 mm window. The assembly passes worst case with 0.02 mm of margin at each end.

That margin is thinner than it looks. It leaves no room for a fourth part, no room for plating thickness, no room for a bearing that arrives at the edge of its own band. It also means the housing tolerance of ±0.10 mm is doing three quarters of the work, so worst case will not let the shop open it up.

Running RSS, and what the margin buys

Root-sum-square of the three tolerances:

√(0.10² + 0.03² + 0.05²) = √0.0134 = 0.116 mm

The predicted spread is 0.30 ± 0.116 mm, or 0.184 to 0.416 mm, against a required 0.10 to 0.50 mm, with 0.084 mm of margin on each side.

Now the useful move. Because that margin is generous, the housing can be loosened from ±0.10 to ±0.15 mm and re-checked:

√(0.15² + 0.03² + 0.05²) = √0.0259 = 0.161 mm

The predicted range becomes 0.139 to 0.461 mm, still inside the window, at roughly 190 ppm of fallout, or about two assemblies per ten thousand.

Worst case with the same housing gives 0.07 to 0.53 mm, failing at both ends. The two methods disagree about whether a cheaper housing is acceptable, and the disagreement is worth money: a ±0.15 mm bore can often be finished in the same pass as roughing, while ±0.10 mm may need a semi-finish pass and a temperature-controlled check.

Which answer is correct depends on what a failure costs. If a bind means a teardown of a finished unit, use worst case. If every unit is test-fitted and a rare tight one is reworked, RSS is defensible and cheaper.

Where RSS breaks down: centring, clocking, datum shift

RSS rests on four assumptions: independence, centring on nominal, approximate normality, and a stated tolerance representing about three sigma.

Each can break in a machine shop. Independence fails when two features are cut in one setup, such as a bore depth and the face it references. Centring fails when a process is deliberately biased, for example a bore targeted high to leave room for a press fit. Normality fails where a physical limit exists, such as a minimum coating thickness. The three-sigma assumption fails whenever a tolerance was set by convention rather than capability, so a ±0.05 mm callout on a feature the process holds to ±0.02 mm is closer to six sigma and RSS overstates its contribution. Many teams multiply the RSS result by 1.5 to cover mean drift; on this example that turns 190 ppm into roughly 2.5%.

Two geometric effects are separate problems. Clocking: a part located on a round boss and a single pin can rotate, and 0.10° over a 120 mm lever arm moves the far end by 0.21 mm, more than every linear tolerance here combined. Every locating scheme needs a feature that prevents rotation. Datum shift: a hole pattern toleranced at MMC gains bonus tolerance from fastener clearance, and the mating part can shift inside it, so the shift must be modelled rather than ignored.

Temperature belongs in the same list. A 200 mm aluminum housing measured at 25 °C and assembled at 20 °C is about 0.011 mm shorter than nominal, using 23 µm/m·K. Small, but not zero when the margin is 0.02 mm.

Writing the result back onto the drawing

A stack-up is only useful if it changes the drawing. The output should be three lists: dimensions that must stay tight because they dominate the variance, dimensions that can be relaxed because their contribution is small, and dimensions that need a datum scheme rather than individual tightening.

Send the stack with your RFQ. When we quote a part like this at JSHS, knowing which dimension drives the assembly tells us where to spend machining effort and where not to, and the price usually falls once that is clear. Send the drawing and the stack and we will quote against the real requirement rather than the worst-case reading of it.

Frequently asked questions

When should I use worst case instead of RSS?

Use worst case when a single failure is unacceptable, when volume is too low for statistics to mean anything, or when you cannot verify the process distribution. Use RSS for production volumes where you can measure the process and detect a rare non-conforming assembly.

Does RSS mean I can ignore the tolerance limits?

No. RSS predicts a distribution, not a guarantee. A dimension can still land at its limit, and that part still has to be reworked or scrapped. RSS changes the expected fallout rate, not the physical limits.

How many parts before RSS is meaningful?

The calculation works from tolerances alone. To validate the assumptions you need enough data to see the distribution, typically 30 parts minimum and 50 to 100 for a stable estimate of spread and centring.

Does a stack-up have to be done in 3D?

Only when the assembly is genuinely three-dimensional, such as a hole pattern with position tolerance in two directions plus a perpendicularity control. For linear axial stacks, a spreadsheet with a sensitivity column is faster and easier to audit than a CAD tolerance model.