Linear Tolerance Stack-Up Worksheet with Worst Case and RSS
A tolerance stack-up answers one question: given the tolerances on every part in a loop, how small and how large can the gap, clearance or interference get? This worksheet handles unequal bilateral tolerances, which are the norm in real drawings, and reports both worst-case and RSS results.
| Item in the loop | Direction (+1 or -1) | Nominal (mm) | Plus tolerance (mm) | Minus tolerance (mm) | Signed nominal (mm) | Mean shift (to tolerance midpoint) (mm) | Half-range (equal bilateral) (mm) | Half-range squared | |
|---|---|---|---|---|---|---|---|---|---|
45 | 0.025 | 0.075 | 0.0056 | ||||||
-12 | 0.06 | 0.06 | 0.0036 | ||||||
-10 | -0 | 0.05 | 0.0025 | ||||||
-15 | -0 | 0.08 | 0.0064 | ||||||
-3 | -0 | 0.02 | 0.0004 | ||||||
-1.5 | 0.015 | 0.045 | 0.002 |
Method: convert every dimension to an equal bilateral form (midpoint plus half-range), sum the midpoints with direction signs for the mean gap, then add half-ranges arithmetically for worst case or in quadrature for RSS. RSS assumes independent, roughly centered, approximately normal distributions with each tolerance spanning about +/- 3 sigma, so it predicts about 99.73 percent of assemblies only under those assumptions. Use worst case for safety-critical fits and low volumes. All units must match.
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How to set up the loop
Draw the assembly gap you care about, then trace a closed path through each part and contact surface. List each dimension as a row. Set the direction to +1 if growing that dimension opens the gap and -1 if it closes it. Enter the nominal and the plus and minus tolerances as positive numbers. The example is an axial end-float stack: a housing depth opens the gap while the bearing, spacer, gear hub, shim and snap ring all close it, leaving a 3.5 mm nominal gap before accounting for unequal tolerances.
- One row per dimension in the loop, no duplicates.
- Use the dimension that actually contacts the next part, not a convenient one.
- Include thermal growth or wear as extra rows if they matter.
- Convert inches or mm consistently before entering.
Worst case versus RSS
Worst case assumes every part is at its limit in the worst direction at the same time, so the band is the arithmetic sum of half-ranges. It guarantees 100 percent interchangeability and often forces tight, expensive tolerances. RSS adds half-ranges in quadrature, the root of the sum of squares, which reflects the low probability of all extremes occurring together. It is only valid when dimensions are independent, centered and roughly normal, and when the process capability behind each tolerance is real. Many aerospace and medical programs use worst case on critical fits and RSS only with documented process capability data.
Reading the example
The example gives a mean gap of about 3.585 mm after shifting each dimension to its tolerance midpoint. The worst-case band is the sum of the half-ranges, 0.375 mm, so the gap ranges from roughly 3.21 to 3.96 mm. The RSS band is about 0.19 mm, giving roughly 3.39 to 3.78 mm. Notice the mean gap differs from the nominal 3.5 because the bearing and housing tolerances are not symmetric. Designing to the nominal instead of the shifted mean is a frequent source of unexpected interference.
Common mistakes
People often forget to shift to the tolerance midpoint when tolerances are unequal, which biases the result. Another is mixing up the direction sign so a closing dimension is treated as opening. Third, applying RSS to a handful of parts from one machine that all share the same offset, which breaks independence. Fourth, ignoring geometric tolerances: a position or perpendicularity callout can add to a 1D loop, so include its worst-case contribution as an extra row when it affects the gap.
Frequently Asked Questions
When should I use RSS instead of worst case?
Use RSS when failure is not safety critical, the assembly volume is high, the dimensions come from independent capable processes, and the tolerances reflect roughly plus or minus three sigma. Use worst case for low volume, safety critical or mandated fits where every assembly must work.
Why is my mean gap different from the nominal?
Because unequal bilateral tolerances move the center of each dimension's range away from its nominal. The sheet shifts every dimension to the midpoint of its tolerance band before summing, which gives the true center of the gap distribution.
Can I include a geometric tolerance in the stack?
Yes. Add a row for the geometric contribution, with its direction sign and a tolerance equal to the worst-case effect on the gap. For example a perpendicularity zone of 0.10 mm adds 0.10 mm of total variation, so enter nominal 0 with plus and minus 0.05 mm.
Does this cover 2D or 3D stacks?
No, this is a one-dimensional loop. Two- and three-dimensional stacks need vector loops with angle sensitivities, or a Monte Carlo simulation. The 1D sheet is still the right first check for axial and radial fits.
If your drawings and BOM live in an ERP or PLM, Netray can help build a private assistant that pulls the dimensions and tolerances for a stack automatically and leaves the engineering decision to you.
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