Tolerance stack-up basics

Eleven short sections, about forty minutes. Nothing here needs the application; every number can be checked with a pencil. Where a section names a sheet or a column, it is the one described in the manual; where it names a formula, the derivation is in the white paper.

1. Why tolerances stack

No dimension is exact. Every one lives somewhere inside its tolerance, and a part is accepted as long as each dimension is inside its own band. Put parts together and the deviations add up: the gap, the wall, the overlap you care about is the sum of several toleranced dimensions, each free to sit anywhere in its band. Stack-up analysis answers one question: for every combination the drawings allow, does the assembly dimension stay within its limits?

2. The chain

The tool is the dimensional chain. Start on one face of the gap, walk through the parts to the other face; every dimension on the way is one link, a contributor. The requirement — the gap itself — closes the loop. Each link has a sign: right (or up) is positive, left (or down) is negative. The requirement has no sign of its own; it is the result. A chain is closed when the signed links bring you back to where the requirement started.

requirement = Σ (sense × nominal)   over the links

One rule keeps chains honest: each link must be a single dimension that exists on one drawing. If you cannot point at the dimension, the link is not a link.

3. Worst case

The arithmetic limits. Every link sits at the extreme that hurts the requirement, all at once. The resulting band is the sum of the tolerances, each end summed on its own, so asymmetric tolerances give an asymmetric band.

MAX = Σ contributions + Σ tol⁺        MIN = Σ contributions − Σ |tol⁻|

Worst case never lies and never happens: with eight links, the chance that all eight are at their worst end at the same time is negligible. It is the band a design is released against, and the only one a customer will sign.

4. RSS

Root-sum-of-squares. If each link varies independently and roughly normally, the variances add, not the tolerances. The band is the square root of the sum of the squared half-tolerances, stated about the chain's mean.

band = √ Σ (tol/2)²    →   result = mean ± band

RSS is convention-free and comparable between tools. It assumes independence and no deliberate bias in the process; when a supplier runs to the high side of every band, RSS is optimistic.

5. Monte Carlo

Simulation. Each link gets a distribution — normal by default, with ±tol taken as ±3σ, or uniform or triangular — and the chain is evaluated hundreds of thousands of times. Out come a histogram, a yield (the share of assemblies inside the limits) and the capability indices.

Cp  = (USL − LSL) / 6σ         Cpk = min(USL − μ, μ − LSL) / 3σ

Cpk 1.33 is a common floor (about 4σ to the nearest limit), 1.67 for critical features. Because the normal distribution is untruncated, a long run occasionally samples beyond ±tol; Monte Carlo can reach slightly past worst case, and that is the convention, not an error.

6. Nominal and mean

A dimension written 10 +0.2/−0 has a nominal of 10 and a band whose middle is 10.1. A chain therefore has two centres. Worst case is stated about the nominal, the way the drawing states its own limits. RSS and Monte Carlo are stated about the mean, because that is what the population is distributed around. For symmetric tolerances the two coincide; for asymmetric ones they differ, and a sheet that prints only one of them is hiding something.

7. Temperature

Parts grow with temperature by their length times the coefficient of thermal expansion times the temperature change. Two materials in one chain grow differently, so a gap that passes at 20 °C can close at 120 °C. The chain is evaluated at the minimum, room and maximum operating temperatures; a nonlinear α(T) curve integrates instead of multiplying. A shrink fit is the same physics used on purpose.

L(T) = L₀ · (1 + α · (T − T₀))

8. Geometric tolerances

A position tolerance on a hole is a zone the hole's axis may wander in; in the chain it enters as a contributor with zero nominal and half the zone as tolerance. When the frame carries Ⓜ, the zone grows as the hole departs from maximum material: the bonus is the difference between the actual size and the maximum-material size. The bonus is zero at maximum material, so the worst-case band becomes asymmetric — a naive sum of absolute tolerances overstates one end.

9. The boundary method

The other school of arithmetic. Instead of size, position and bonus as three links, compose the inner and outer boundaries of a feature — the envelopes no material may cross — and use them as one contributor. Both schools agree on the worst-case limits when applied correctly; the boundary method is shorter for fastener assemblies, the contributor method is more transparent when someone asks which tolerance to tighten.

10. A worked example

Three blocks in a housing, a published case. The housing interior is 4.020 ±0.005; the blocks are 2.000 ±0.005, 1.500 ±0.003 and 0.500 ±0.002 (inches). The requirement is the end gap, which must stay between 0 and 0.050.

worst case   tol = 0.005 + 0.005 + 0.003 + 0.002 = 0.015   →   0.005 … 0.035   PASS
RSS          band = √(0.005² + 0.005² + 0.003² + 0.002²) = 0.0079   →   0.012 … 0.028
Monte Carlo  σ ≈ 0.0026 (±tol = ±3σ)  →  yield ≈ 100 %, Cpk ≈ (0.020 − 0) / (3 · 0.0026) ≈ 2.5

The chain closes: +4.020 − 2.000 − 1.500 − 0.500 = +0.020, the gap on the drawing. The worst-case band sits inside 0 … 0.050 with 0.005 to spare at the tight end. RSS says the population will use about half of that band. Both are true; they answer different questions.

11. Exercise

Tighten Block A to ±0.003 and loosen the housing to ±0.008. Recompute worst case and RSS by hand. Then ask the question that stack-up is really for: which single tolerance, if loosened, buys the most cost for the least risk? The RSS variance shares answer it — the link with the largest share is the one to tighten, and the ones with the smallest shares are the ones to loosen. The application prints those shares as a Pareto in the Results sheet; the 30-day trial is enough to try it on your own assembly.