Tolerance stacks — AlpineTutor

Every part in an assembly is in tolerance. The assembly does not go together. This topic is about finding that out with a pencil instead of on the line.

A partly assembled transmission housing on a clean assembly bench with its stack of shims, spacers and two tapered roller bearings laid out in sequence beside the open face, an assembler's gloved hands at the edge of the frame and soft window light from the left. — Every shim in that row is a contributor. So is the housing, the bearing width and the shoulder it seats against — and the only thing the assembly actually cares about is the gap left at the end of the chain.

The loop

A stack is a closed chain of dimensions, starting at one face of the gap you care about, running through the parts by way of features that genuinely locate one another, and returning to the other face of the gap.

Two rules decide whether the answer means anything:

The loop must close. Start and finish at the two faces forming the gap.
It must pass only through real contacting features. A dimension that nothing actually locates from does not belong in the chain, however neatly it is drawn.

Two ways to add up

A comparison of worst-case and root-sum-square tolerance stacking for the same four-part assembly. The left panel adds four tolerances of plus or minus 0.1 millimetres arithmetically to give plus or minus 0.4. The right panel combines them as the square root of the sum of squares to give plus or minus 0.2. Below, two distribution curves are overlaid on the same gap limits, showing that the worst-case span covers the full range while the statistical distribution concentrates near nominal, with the tails that the statistical method accepts marked in warning colour. — The same four parts, added two different ways. Worst case guarantees fit and costs tolerance. Root sum square recovers the tolerance and accepts a small tail — which is only honest if the assumptions behind it hold.

Worst case adds every tolerance in the loop:

4 contributors at ± 0.1
worst case  =  0.1 + 0.1 + 0.1 + 0.1  =  ± 0.4

No assembly can ever fail. It is an arithmetic guarantee, and that is why it is used on safety-critical items, low-volume builds and anything where a single failure is unacceptable.

Root sum square assumes the contributors vary independently and rarely conspire:

RSS  =  √(0.1² + 0.1² + 0.1² + 0.1²)  =  √0.04  =  ± 0.2

Half the stack, for nothing but a change of assumption. On a high-volume assembly that is usually the difference between a design that can be made and one that cannot.

Where the tolerance actually goes

Lever	Typical effect	Cost
Remove a part from the loop	Removes a whole contributor permanently	Usually the cheapest of all
Attack the two largest contributors	Most of the gain, small change	Low
Tighten everything proportionally	Same gain, every part more expensive	High
Switch worst case to RSS	Halves the number on paper	Free, and sometimes dishonest

Two things that are usually left out and should not be

Bonus tolerance. The bonus from a maximum material condition modifier appears only as a feature departs from maximum material — and the worst case is defined at maximum material, where the bonus is zero. Counting it in a worst-case stack claims tolerance that is absent exactly when it is needed.

Temperature. An aluminium housing on a steel shaft through a hundred-degree rise moves the gap by more than the entire manufacturing spread. A stack computed at 20 °C for an assembly that runs hot has answered a question nobody asked.

What to take away

A stack is a closed loop through genuinely contacting features. If it does not close, it describes a different assembly.
Worst case adds tolerances and guarantees fit. Use it for safety-critical and low volume.
Root sum square combines them as √(Σt²) and recovers roughly half — when the contributors are independent, centred and numerous.
Correlated tools, off-centre processes and small builds all break the RSS assumptions quietly.
Removing a part from the loop beats tightening any tolerance in it.
Exclude bonus tolerance from a worst-case stack; it is zero where the worst case lives.
Include thermal effects on anything that runs hot or mixes materials.
A negative minimum gap means some in-tolerance parts will not assemble. Believe it.