Tolerance Stack-up — validation & limitations

What it calculates

Engineering estimate — this tool uses an accepted simplified model rather than one citable governing standard. Use it for preliminary sizing and verify the final design against manufacturer data or a qualified engineer.

• Worst-case (WC) tolerance stack-up: guaranteed 100 % interchangeability for any production quantity, with signed links and asymmetric limits (+upper/−lower) rather than ±tolerance only
• RSS statistical stack-up: closing tolerance scales as √N, quantifying the relaxation available over large batch sizes
• ISO 286-1/-2:2010 shaft/bore fit analysis — clearance, transition, and interference — with H7/h6, H7/g6, H7/f7, H9/d9, H7/k6, H7/m6, H7/n6 and H7/p6 read from the published tables at any basic size up to 500 mm
• Capability of the closing gap against your required minimum clearance (Gap Cpk), with a shortfall-rate estimate — plus a standalone Cp/Cpk panel that takes your own specification limits and measured process sigma
• Monte Carlo simulation (1 000 samples, Box-Muller RNG) of the assembled gap distribution
• Per-application fit recommendations (bearing inner/outer rings, gear hubs, press fits, sliding fits) and every offered fit compared at your basic size, with interface pressure and hub hoop stress for the interference classes

Standards this tool applies

• ASME Y14.5 — 1-D worst-case and RSS tolerance chain analysis plus ISO 286-1/-2 shaft/bore fit analysis — used to set and verify the positional tolerance budget that feeds GD&T callouts.
• ISO 14253-1 — 1-D worst-case and RSS chain analysis, ISO 286-1/-2 shaft/bore fits, capability indices and Monte Carlo simulation — the tolerance budget that a conformance decision is later taken against. It does NOT apply a measurement-uncertainty guardband; that decision is available through the GD&T API only.

Validation evidence — independent reference cases (3)

Each case runs the tool's engine on a worked example whose values come from a published source or a hand derivation from the cited equations, and an automated regression test asserts the engine against those values within the stated tolerance. "Conservative" means the engine is known to sit on the safe side of the reference and the test asserts that side. Sources are cited; their text is not reproduced.

• ISO 286 fit limits at 40 mm and interference pressure

  Agrees

  Source: ISO 286-1:2010 §5 / Annex A; Shigley's Mechanical Engineering Design, 10th ed. §3-16, eq. 3-56/3-58

  Inputs: Dia 40 mm: H7/g6, H8/f7, H7/k6; steel E 210 GPa, hub OD/bore 2

  Quantity	Reference value	Tolerance
  H7/g6 clearance range (um)	+9 to +50	exact
  H7/k6 clearance range (um)	-18 to +23	exact
  contact pressure H7/k6 (MPa)	35.4375	1e-9 relative
  hub hoop stress (MPa)	59.0625	1e-9 relative

  tests/golden/REF-tolerances-iso286-fit.golden.test.ts

• Process capability Cp, Cpk and nonconforming ppm

  Conservative

  Source: Montgomery, Introduction to Statistical Quality Control (6th+ ed.), §8.3, Table 8.2; ISO 22514-2 indices

  Inputs: USL 10.05, LSL 9.95; centred mean 10.00, sigma 0.0125; off-centre mean 10.01, sigma 0.010

  Quantity	Reference value	Engine	Tolerance
  Cp / Cpk centred	1.333333	—	1e-9
  ppm centred	63.3425	—	0.1 ppm absolute
  Cp / Cpk off-centre	1.666667 / 1.333333	—	1e-9
  ppm off-centre (true fraction)	31.672	63.3425	one-sided (engine >= true fraction), 0.1 ppm

  Conservative simplification: The engine converts Cpk to a two-tailed 2·Phi(-3·Cpk) figure, so the off-centre case reads 63.34 ppm against a true nonconforming fraction of 31.67 ppm, about twice as high.

  tests/golden/REF-tolerances-process-capability.golden.test.ts

• 1-D tolerance chain worst case, RSS and gap capability

  Agrees

  Source: Fortini, Dimensioning for Interchangeable Manufacture (1967); Drake, Dimensioning and Tolerancing Handbook (1999) ch. 9; ASME Y14.5; Montgomery, Introduction to Statistical Quality Control §8.2

  Inputs: Chain +50±0.10, -20±0.05, -25±0.08, -4.5±0.03, gap >= 0.300 mm; asymmetric slot 20 +0.1/0 vs block 19.9 0/-0.1, g_min 0.120

  Quantity	Reference value	Tolerance
  T_WC / T_RSS (mm)	0.26 / 0.1407125	1e-9
  G_min RSS (mm)	0.3592875	1e-9
  gap Cpk	1.421338	1e-8
  shortfall ppm	10.0393	0.1 ppm absolute

  tests/golden/REF-tolerances-stackup.golden.test.ts

Limitations

Conservative simplifications

• Process capability Cp, Cpk and nonconforming ppm — The engine converts Cpk to a two-tailed 2·Phi(-3·Cpk) figure, so the off-centre case reads 63.34 ppm against a true nonconforming fraction of 31.67 ppm, about twice as high.

Not checked by this tool

• Geometric tolerances (form, orientation, position) in the stack — 1-D linear stack only; add GD&T zones as extra links or use /gdt
• Angular and 2-D / 3-D stacks — All links act along one axis with sensitivity 1; use a vector-loop or 3-D tolerance analysis
• Thermal expansion of the parts in service — Dimensions at the reference temperature; add the differential growth as a link
• Correlated links (same machine, fixture or setup) in the RSS result — RSS assumes independent links; use measured data or treat correlated links worst-case
• Fit: hub stress and press-in force — The interface pressure is an estimate (steel, k = 2); use /pressfit for the joint
• Fit: form errors (ovality, taper) and roughness smoothing — Perfect form is assumed; add form tolerances and roughness loss (DIN 7190-1)
• Measurement uncertainty at inspection (ISO 14253-1) — Not applied here; use guard bands for the gauge uncertainty at inspection