Tolerance Stack-up for Operators
Add up part tolerances by the worst-case and RSS methods, read a stack-up report, find the contributor that matters most, and spot when process drift breaks the math.
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Parts that each measure in spec can still assemble into something that does not fit. Small errors add up along a chain of parts, and that is tolerance stack-up. It happens with bearings and spacers on a shaft, panels in a frame, brackets on a welded rail, or any row of parts that has to land inside a limit. If you understand it, you can see a problem on paper before it shows up at assembly.
What You Will Be Able to Do
- Work out a worst-case stack and an RSS stack by hand.
- Say when each method is the right one.
- Read a stack-up report and find the biggest contributor.
- Name the assumptions behind RSS.
- Recognize the process signals that mean the stack is no longer safe.
Why Stack-up Matters
Take three spacers on a shaft between two bearings. Each is 1.000 long with a tolerance of +/- 0.005.
- Nominal stack: 1.000 + 1.000 + 1.000 = 3.000.
- All three at maximum: 1.005 x 3 = 3.015.
- All three at minimum: 0.995 x 3 = 2.985.
If the assembly must be 3.000 +/- 0.008, then three good spacers can miss by up to 0.015, almost double the limit. Every part passed, and the assembly can still fail. Engineering then has to tighten the spacers, loosen the assembly limit, or accept a known risk. Which choice makes sense depends on the method used to add the tolerances.
Worst-Case Stack
The worst-case method adds every tolerance directly. It assumes every part lands at its worst extreme at the same time.
For the three spacers: 0.005 + 0.005 + 0.005 = +/- 0.015.
If the assembly limit is wider than the worst-case stack, no combination of in-spec parts can fail. That guarantee costs money, because it pushes part tolerances tight or assembly limits wide. It is the right choice for one-off builds, safety-critical assemblies and small quantities. In high-volume work it is often more than the job needs, because all the parts rarely hit their extremes together.
Root-Sum-Square Stack
The root-sum-square (RSS) method is statistical. Parts vary at random, so highs and lows tend to cancel. Square each tolerance, add the squares, and take the square root.
For the three spacers:
- Square each: 0.005 x 0.005 = 0.000025.
- Add them: 0.000025 x 3 = 0.000075.
- Take the square root: 0.00866.
The RSS stack is +/- 0.0087, a bit over half of the worst-case 0.015. If each tolerance equals about three standard deviations of its process, about 99.73 percent of assemblies land inside that range. Here that range is still slightly wider than the +/- 0.008 limit, so RSS predicts roughly 1 assembly in 200 outside it. That is a business decision, and it belongs to engineering.
RSS only works when these things are true:
- Each dimension follows a bell-shaped distribution.
- Each dimension varies independently of the others.
- Each process is centered on its nominal.
- Each tolerance is about three standard deviations wide, or wider.
When these break down, RSS under-predicts the real stack, and more assemblies fail than the math said. Real production usually falls between the RSS answer and the worst case.
Worked Example: Three Parts End to End
| Part | Nominal | Tolerance | Tolerance squared |
|---|---|---|---|
| A | 1.000 | +/- 0.005 | 0.000025 |
| B | 2.000 | +/- 0.010 | 0.000100 |
| C | 1.500 | +/- 0.005 | 0.000025 |
| Total | 4.500 | +/- 0.020 worst case | 0.000150 |
- Worst case: 4.500 +/- 0.020, so 4.480 to 4.520.
- RSS: the square root of 0.000150 is 0.01225, so 4.48775 to 4.51225.
RSS comes to about 60 percent of worst case. Part B dominates. Its squared tolerance is four times the size of A's or C's, because squaring punishes the largest number. If engineering wants a tighter assembly, tightening B does far more than tightening A or C.
Reading a Stack-up Report
Most reports chase a gap or a fit along a chain of parts. On a critical assembly, ask whether engineering has one, and read it if they do. Here is one for the gap left in a housing after three spacers and a shim go in.
| Contributor | Nominal | Tolerance | Sensitivity |
|---|---|---|---|
| Housing depth | 3.020 | +/- 0.002 | +1 |
| Spacer 1 | 1.000 | +/- 0.005 | -1 |
| Spacer 2 | 1.000 | +/- 0.005 | -1 |
| Spacer 3 | 1.000 | +/- 0.005 | -1 |
| Shim | 0.010 | +/- 0.003 | -1 |
The report's results work out like this:
| Result | Value | Gap range |
|---|---|---|
| Nominal gap | 3.020 - 3.010 = 0.010 | Not a range |
| Worst case | +/- 0.020 | -0.010 to 0.030 |
| RSS | +/- 0.0094 | 0.0006 to 0.0194 |
Sensitivity says how much the result moves when that dimension moves. +1 adds one for one. -1 subtracts one for one, so a longer spacer closes the gap. A value of 2 counts double, and 0.1 barely matters. Each contributor's share is its sensitivity times its tolerance, and those shares are what get added or squared.
Read the numbers. The worst case goes negative, which means parts could bind. RSS says that is unlikely if the assumptions hold. That is exactly where someone needs to decide which risk is acceptable.
Which Contributor Matters Most
Ranking contributors tells you where effort pays off. In the report above, the sum of squares is 0.000088.
| Contributor | Squared share | Percent of total |
|---|---|---|
| Each spacer | 0.000025 | 28 percent |
| Shim | 0.000009 | 10 percent |
| Housing depth | 0.000004 | 5 percent |
The three spacers together make up about 85 percent. Tightening the housing would cost money and change almost nothing. Knowing the top two or three contributors tells a shop which tolerances to tighten, which could be loosened, and where inspection and charting time should go.
Position Tolerances and Basic Dimensions in a Stack
A boxed basic dimension is exact, so it adds nothing to the stack. The tolerance lives in the feature control frame. A position zone of Ø 0.010 lets the axis wander 0.005 in any direction, so it counts as +/- 0.005 in a linear stack. Use half the diameter, not all of it.
Stacks in More Than One Direction
- 1D (linear): dimensions along one axis, as in the examples here. RSS applies cleanly and you can do these by hand.
- 2D (planar): hole locations in an X-Y plane or bearings that must line up. Position tolerances split into X and Y parts using basic trigonometry.
- 3D (spatial): features on several faces, or angles that matter. These usually need software.
Software tools use Monte Carlo simulation. The program builds thousands of virtual assemblies from random parts and reports how the final dimension comes out. It handles skewed distributions, angles, unequal tolerances and features that drift together, such as two cut in one setup. Engineering runs it. You see the result as tighter or looser tolerances on the print. If you meet a 3D problem, hand it to engineering.
When Process Drift Breaks RSS
RSS assumes each process is centered. Suppose a process runs 0.003 above nominal with a tolerance of +/- 0.005. Every part measures in spec, so inspection passes.
Stack three of those parts and the mean of the stack is 3 x 0.003 = 0.009 above nominal. The RSS range around that shifted mean is still +/- 0.0087, so real stacks run from about 0.0003 above nominal to about 0.0177 above nominal. Nothing is centered, and the high side now reaches more than double the +/- 0.008 limit from the first example. Every individual part passed.
That is why control charts track centering as well as spread. A process with good spread but poor centering shows it as a Cp well above its Cpk, and that process breaks the RSS assumptions even when inspection is clean.
Watch for these signals on a critical dimension, and act on them as your control plan directs:
- The chart for that dimension signals a shift or a trend, even though the parts are in spec.
- The capability figure falls below the plan's target. A falling Cpk means the process is eating its margin.
- Assembly starts reporting tight fits or loose fits that inspection did not predict. That can mean two contributors are drifting together or a contributor is off center.
Stopping to recenter costs hours. Field returns from a stack that went bad cost far more. Do not adjust on your own judgment, though. Follow the reaction plan, and tell the lead what you saw.
Common Mistakes
| Mistake | What it costs |
|---|---|
| Assuming in-spec parts always assemble | Failed builds, and a hunt for a bad part that does not exist |
| Using RSS on a handful of parts | Too few parts for the odds to work, so failures run high |
| Adding a whole diameter zone to a stack | The stack looks twice as bad as it is |
| Getting the sign wrong in a gap | A wrong answer on whether parts fit |
| Tightening every tolerance equally | Cost goes up with little effect |
| Treating a drifted mean as fine because parts pass | Assemblies fail with no warning at inspection |
Check Your Understanding
1. Four parts each carry +/- 0.004. What are the worst-case and RSS stacks?
Worst case is 4 x 0.004 = +/- 0.016. RSS is the square root of 4 x 0.000016, which is the square root of 0.000064, or +/- 0.008.
2. A one-time test rig has three parts in a chain, and it must fit. Which method should you use?
Worst case. With so few parts there is no large population for the odds to average out, and a quick worst-case sum gives you an upper limit you can trust.
3. A stack has contributors of +/- 0.002, +/- 0.003 and +/- 0.012. Which should get the attention?
The +/- 0.012 contributor. Its square is 0.000144, against 0.000004 and 0.000009 for the others, so it drives nearly the whole result.
4. Every part in a run passes, but the chart shows the mean creeping up, and the assembly line reports tight fits. What is happening?
The process is off center, so stacked parts add their offset together. Inspection only sees that each part is in spec. Report it and follow the reaction plan to recenter.
Key Takeaways
- In-spec parts can still stack into an out-of-spec assembly.
- Worst case adds all tolerances. It is a safe upper limit and can be expensive to meet.
- RSS squares, adds and takes the root. It needs centered, independent, bell-shaped processes.
- The largest tolerance usually dominates, so find it first.
- A drifting mean breaks RSS even when every part passes. Watch the chart as well as the parts.
- Stacks with angles or features on several faces belong to engineering and its software.