1D linear chain · Worst-case · RSS · Shaft-bore clearance / interference
Tolerance Stack-up Calculator — Worst-Case, RSS & ISO 286 Shaft/Bore Fits
Governing standard: ISO 286· ISO 286-1/-2:2010 (limits and fits) — implemented from the published tables. Worst-case and RSS chain combination are established engineering practice, not clauses of a standard; capability indices follow ISO 22514-2.
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The MechanixCalc tolerance stack-up calculator analyses 1D linear dimension chains by the two standard engineering methods — worst case and RSS. Enter each link in the chain — its nominal dimension, its upper and lower tolerance limits (equal for a plain ±tolerance, different for an asymmetric band such as +0.050/0), and whether it opens or closes the gap — together with the minimum clearance the assembly must hold, and the tool returns the worst-case and RSS closing gap, the capability of that gap against your requirement, and a Monte Carlo simulation of the assembled distribution.
It is built for mechanical design, metrology, and manufacturing engineers who need to verify that a multi-component assembly will always maintain the required clearance or interference. The integrated ISO 286-1/-2 fit analyser reads the published limit deviations at your actual basic size — they widen with diameter, so IT7 is 10 µm at ⌀3 mm and 63 µm at ⌀500 mm — and covers clearance, transition, and interference fits with application recommendations for bearings, gear hubs, press fits, and sliding assemblies.
What this calculator does
- 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
- Branded PDF engineering report with the full method, input summary, and results
Method & formulas
Worst-case and RSS tolerance stack-up
No international standard normatively defines how a 1-D tolerance chain is combined; worst case and RSS are established engineering practice, set out in Fischer's Mechanical Tolerance Stackup and Analysis and Drake's Dimensioning and Tolerancing Handbook. (ISO 14253-1, sometimes cited here, governs something different: the accept/reject decision under measurement uncertainty. This tool applies no guardband and does not claim it.) The worst-case method sums the tolerance half-bands arithmetically to guarantee that every assembly in any production run meets the closing dimension. The RSS method treats each dimension as a statistically independent random variable — normal, centred on nominal, with ±tolerance taken as ±3σ — and combines the standard deviations in quadrature. The RSS closing tolerance grows as √N instead of N, so it is far tighter for long chains but depends entirely on those assumptions holding: links machined on a shared setup are correlated, and the RSS bound then understates the real spread.
For each link the closing gap is computed from the signed sum of the nominal dimensions (positive for dimensions that open the gap, negative for those that close it), and the WC and RSS tolerances are then added and subtracted to bound the gap range.
G_nom = Σ (L_i · s_i)where G_nom = nominal closing gap; L_i = nominal length of link i; s_i = +1 for a positive (opening) link, −1 for a negative (closing) link
T_WC = Σ t_i → G ∈ [G_nom − T_WC, G_nom + T_WC]where T_WC = worst-case accumulated tolerance; t_i = bilateral tolerance of link i (half-width)
T_RSS = √(Σ t_i²) → G_RSS ∈ [G_nom − T_RSS, G_nom + T_RSS]where T_RSS = RSS closing tolerance; t_i = bilateral tolerance of link i, treated as ±3σ_i so σ_i = t_i / 3
Capability of the closing gap (Gap Cpk)
A capability index is only meaningful against a specification set INDEPENDENTLY of the process. The closing gap's specification is the minimum clearance the assembly must hold, which you enter, so the tool reports the one-sided index Gap Cpk = (G_nom − G_req) / (3 · σ_RSS) with σ_RSS = √Σ(t_i/3)². It falls below 1.33 when the design has little statistical room and goes negative when the nominal gap itself does not meet the requirement. Gap Cpk ≥ 1.33 is the usual minimum for general manufacturing and ≥ 1.67 the target for safety-critical assemblies.
Gap Cpk = (G_nom − G_req) / (3 · σ_RSS)where G_nom = nominal closing gap; G_req = minimum clearance the assembly must hold; σ_RSS = √Σ(t_i / 3)² = assembly standard deviation, with each half-band ASSUMED to be 3σ. This is a design-stage allocation aid modelled on the ISO 22514-2 definition, not a capability study: a true index needs σ estimated from measured data on a process in statistical control.
T_WC / T_RSS = Σ t_i / √(Σ t_i²)where How much tolerance RSS buys back over worst case. It is fixed by the SHAPE of the tolerance set alone — it equals √N for N equal tolerances and can never fall below 1 — and reads no nominal dimension, so it says nothing about whether the assembly fits.
ISO 286 shaft/bore fit analysis
ISO 286-1/-2:2010 defines fundamental deviations and standard tolerance grades for cylindrical fits, both as functions of the basic size: the standard tolerance factor is i = 0.45·∛D + 0.001·D micrometres with D the geometric mean of the diameter band, and the clearance-shaft deviations follow their own power laws (d: −16·D^0.44, f: −5.5·D^0.41, g: −2.5·D^0.34). Selecting a fit therefore reads the published table AT YOUR BASIC SIZE — H7/g6 gives 2 to 18 µm of clearance at ⌀3 mm and 20 to 123 µm at ⌀500 mm. You can also enter the four deviations by hand. The tool returns the maximum and minimum clearance (positive) or interference (negative) and classifies the fit. For interference fits it estimates the Lamé interface pressure and the resulting hub hoop stress for a solid steel shaft in a like-material hub with an assumed outer-to-bore ratio k = 2 — an engineering estimate; use the press-fit calculator for real hub geometry or dissimilar materials.
C_max = D_max − d_min = (D + ES) − (d + ei); C_min = D_min − d_max = (D + EI) − (d + es)where C_max = maximum clearance (positive) or maximum material interference (negative); D = bore nominal; d = shaft nominal; ES, EI = bore upper/lower deviations; es, ei = shaft upper/lower deviations
p = (E · δ / d) · (k² − 1) / (2k²)where p = contact pressure (MPa); E = 210 000 MPa (steel); δ = interference (mm) = |C_min| when C_min < 0; d = nominal diameter (mm); k = 2 (hub outer-to-bore diameter ratio assumed)
Worked example
Three-component assembly: housing bore depth A1 = 30 mm (+, opens the gap), spacer A2 = 12 mm (−, closes it), shaft shoulder A3 = 15 mm (−, closes it). Tolerances: t1 = ±0.06 mm, t2 = ±0.04 mm, t3 = ±0.03 mm. The joint must retain at least 2.80 mm of running clearance. Find the worst-case and RSS closing gap and the capability of that gap against the requirement.
Given
- A1 (housing bore depth, + direction)30.000 mm, t = ±0.06 mm
- A2 (spacer, − direction)12.000 mm, t = ±0.04 mm
- A3 (shaft shoulder, − direction)15.000 mm, t = ±0.03 mm
- Required minimum gap G_req2.800 mm
Result
- Nominal gap G_nom3.000 mm
- Worst-case gap range[2.870, 3.130] mm
- RSS gap range (±3σ)[2.9219, 3.0781] mm
- RSS tolerance T_RSS0.078102 mm
- Gap Cpk vs the 2.800 mm requirement2.561 (Excellent)
- Worst-case-to-RSS ratio1.664
- VerdictPASS — worst-case minimum gap 2.870 mm clears the requirement
- Nominal closing gap: G_nom = 30 − 12 − 15 = 3.000 mm.
- Worst-case accumulated tolerance: T_WC = 0.06 + 0.04 + 0.03 = 0.13 mm.
- WC gap range: [3.000 − 0.13, 3.000 + 0.13] = [2.870, 3.130] mm. Minimum WC gap 2.870 mm ≥ 2.800 mm — PASS, with 0.070 mm to spare, so every combination of in-tolerance parts holds the clearance.
- RSS tolerance: T_RSS = √(0.06² + 0.04² + 0.03²) = √(0.0036 + 0.0016 + 0.0009) = √0.0061 = 0.078102 mm.
- RSS gap range (±3σ): [3.000 − 0.078102, 3.000 + 0.078102] = [2.9219, 3.0781] mm.
- Assembly standard deviation: σ_RSS = T_RSS / 3 = 0.078102 / 3 = 0.026034 mm.
- Capability of the gap: Gap Cpk = (G_nom − G_req) / (3 · σ_RSS) = (3.000 − 2.800) / (3 × 0.026034) = 0.200 / 0.078102 = 2.561 — Excellent, comfortably past both the 1.33 general-manufacturing and 1.67 safety-critical thresholds.
- Worst-case-to-RSS ratio: T_WC / T_RSS = 0.13 / 0.078102 = 1.664. That is how much tolerance RSS would buy back — it is fixed by the shape of the tolerance set, not by the nominals, so it is reported separately and is not a capability index.
Illustrative example — verify against your actual nominal dimensions, tolerances, and assembly directions. The worst-case minimum of 2.870 mm is a 100 % guarantee for parts made within tolerance; the Gap Cpk of 2.561 additionally assumes each half-band is a centred ±3σ spread of an independent process, which is a design-stage convention rather than a measured capability study.
Frequently asked questions
Which standard does this tolerance stack-up calculator use?
One standard is implemented and it is ISO 286-1/-2:2010 — the shaft/bore limit deviations come from its published tables, read at your basic size, with no scaling or interpolation between bands. The worst-case and RSS chain-combination rules are NOT clauses of any international standard: no ISO or ASME document defines them, and they are established engineering practice (Fischer, Mechanical Tolerance Stackup and Analysis; Drake, Dimensioning and Tolerancing Handbook). Capability indices follow the ISO 22514-2 definition. ISO 14253-1:2017 is a different thing again — it governs the accept/reject decision under measurement uncertainty, and its U95 guardband is implemented in the MechanixCalc GD&T calculator, not here.
What is the difference between worst-case and RSS tolerance analysis?
Worst-case (WC) analysis sums all tolerances arithmetically and guarantees 100 % interchangeability regardless of how many parts are made or where they source from. It is mandatory for safety-critical or very low-volume assemblies. RSS (root-sum-square) analysis treats each dimension as statistically independent and accumulates the standard deviations in quadrature (T_RSS = √Σt_i²). The RSS closing tolerance grows as √N rather than N, so it allows looser individual tolerances for the same assembly yield — but only if the process is centred and the statistical assumptions hold.
When should I use worst-case versus RSS tolerance stack-up?
Use worst-case analysis when: (a) production volume is low (< ~50 assemblies), (b) parts come from multiple unqualified suppliers, (c) the assembly is safety-critical and 100 % interchangeability is non-negotiable, or (d) the individual tolerances are wide relative to the closing gap. Use RSS when production volumes are large, the process is statistically controlled and centred, and there is an acceptable non-zero risk of rejects — and always pair it with a Gap Cpk ≥ 1.33 check against the clearance the assembly actually has to hold.
How does the ISO 286 fit analyser work?
ISO 286 defines a system of fundamental deviations (letters a–zc for shafts, A–ZC for bores) and standard tolerance grades (IT01–IT18), both of which are functions of the basic size. Pick a standard H-basis pair — H7/h6 close clearance, H7/g6 or H7/f7 running, H9/d9 loose, H7/k6, H7/m6 or H7/n6 transition, H7/p6 press — and the calculator reads the published limit deviations from the ISO 286-1 table at your basic size, or you can enter the four deviations by hand. This matters: H7/g6 gives 2 to 18 µm of clearance at ⌀3 mm and 20 to 123 µm at ⌀500 mm, so the designation alone does not fix a clearance. The tool returns the maximum and minimum clearance, classifies the fit type, and — for interference fits — estimates the Lamé interface pressure and the hub hoop stress for a solid steel shaft in a like-material hub with an assumed outer-to-bore ratio k = 2. Tabulated for 0 < d ≤ 500 mm, H-basis holes; outside that range the tool says so rather than extrapolating.
Is the tolerance stack-up calculator free?
Yes, this tool is completely free. You can use it immediately with no sign-up or account required. A free 14-day account trial (no credit card needed) unlocks saved calculations across all MechanixCalc tools. The branded PDF engineering report and persistent cloud save are included in the free 14-day trial and in every paid plan.
Related calculators
- GD&T CalculatorApply geometric dimensioning and tolerancing (form, orientation, position) alongside dimensional stack-up.
- Press Fit Analysis (DIN 7190)Full Lamé press-fit sizing — contact pressure, assembly force, and fatigue — once the ISO 286 fit is selected.
- Shaft AnalysisVerify the shaft under fatigue and deflection once the fits and tolerances are established.
- Bearing Analysis (ISO 281)Confirm bearing L10 life using the same fit parameters — bearing inner-ring interference fit directly feeds radial preload.
- Machining ParametersCalculate the cutting parameters and surface finish required to achieve the specified dimensional tolerances.
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