1D linear chain · Worst-case · RSS · Shaft-bore clearance / interference
Tolerance Stack-up Calculator — Worst-Case, RSS & ISO 286 Shaft/Bore Fits (ISO 14253-1)
Governing standard: ISO 14253-1· ISO 14253-1:2017 (verification by measurement) · ISO 286-1/-2:2010 (limits and fits) · Six Sigma Cp/Cpk
The MechanixCalc tolerance stack-up calculator analyses 1D linear dimension chains to ISO 14253-1, the international standard for verification of workpieces by measurement. Enter each link in the chain — its nominal dimension, bilateral tolerance, and direction — and the tool instantly returns the worst-case and RSS (root-sum-square) closing gap, process capability indices Cp and Cpk, and a Monte Carlo simulation of the assembled gap 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 — and who need a standards-cited, auditable calculation rather than a back-of-envelope estimate. The integrated ISO 286 fit analyser covers clearance, transition, and interference fits for shaft/bore pairs, with application recommendations for bearings, gear hubs, press fits, and sliding assemblies.
What this calculator does
- ISO 14253-1 worst-case (WC) tolerance stack-up: guaranteed 100 % interchangeability for any production quantity
- RSS statistical stack-up: closing tolerance scales as √N, quantifying the relaxation available over large batch sizes
- ISO 286 shaft/bore fit analysis — clearance, transition, and interference — with built-in H7/h6, H7/g6, H7/k6, H7/p6, and other standard pairs
- Six Sigma process capability indices Cp and Cpk with PPM defect estimate and sigma-level display
- Monte Carlo simulation (1 000 samples, Box-Muller RNG) of the assembled gap distribution
- ISO fit selection guide with per-application recommendations (bearing inner/outer rings, gear hubs, press fits, sliding fits)
- Branded PDF engineering report with the full method, input summary, and results
Method & formulas
Worst-case and RSS tolerance stack-up (ISO 14253-1)
ISO 14253-1 defines the verification decision rule for workpieces and measuring equipment: the worst-case (WC) method sums all individual tolerances 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 — assuming a normal distribution centred on the nominal with ±tolerance = ±3σ — and combines the standard deviations in quadrature. The RSS closing tolerance grows as √N instead of N, so it is significantly tighter for long chains but relies on the statistical assumptions being met.
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
Six Sigma process capability (Cp / Cpk)
The process capability index Cp quantifies how much headroom the process has relative to the specification. The calculator uses the one-sided convention: Cp = T_WC / (6 · σ_RSS), where T_WC is the worst-case half-band (Σ t_i) and σ_RSS = √Σ(t_i / 3)². Cp ≥ 1.33 is the minimum for general manufacturing; Cp ≥ 1.67 is the target for safety-critical assemblies. Because σ_RSS = T_RSS / 3 and T_RSS ≤ T_WC, the Cp value reflects how much statistical margin exists inside the worst-case envelope.
Cp = T_WC / (6 · σ_RSS)where Cp = process capability index; T_WC = worst-case closing tolerance (half-band, i.e. one-sided accumulation Σ t_i); σ_RSS = √Σ(t_i / 3)² = assembly process standard deviation
ISO 286 shaft/bore fit analysis
ISO 286-1/-2 defines fundamental deviations and tolerance grades for cylindrical fits. The calculator accepts the shaft upper and lower deviations (es, ei) and the bore upper and lower deviations (ES, EI) and returns the maximum and minimum clearance (positive) or interference (negative). For interference fits, the Lamé contact pressure for a steel solid shaft in a like-material hub (outer-to-inner diameter ratio k = 2 assumed) is also estimated.
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 A1 = 30 mm (+, opens gap), spacer A2 = 12 mm (−, closes gap), shaft shoulder A3 = 15 mm (−, closes gap). Tolerances: t1 = ±0.06 mm, t2 = ±0.04 mm, t3 = ±0.03 mm. Required minimum gap ≥ 0. Find the worst-case and RSS closing gap.
Given
- A1 (housing, +direction)30.000 mm, t = ±0.06 mm
- A2 (spacer, −direction)12.000 mm, t = ±0.04 mm
- A3 (shoulder, −direction)15.000 mm, t = ±0.03 mm
Result
- Nominal gap G_nom3.000 mm
- Worst-case gap range[2.870, 3.130] mm
- RSS gap range (±3σ)[2.922, 3.078] mm
- RSS tolerance T_RSS0.0781 mm
- Assembly Cp≈ 0.83
- 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 > 0 — PASS.
- RSS tolerance: T_RSS = √(0.06² + 0.04² + 0.03²) = √(0.0036 + 0.0016 + 0.0009) = √0.0061 ≈ 0.0781 mm.
- RSS gap range: [3.000 − 0.0781, 3.000 + 0.0781] = [2.922, 3.078] mm.
- Process standard deviation: σ_RSS = T_RSS / 3 ≈ 0.0781 / 3 ≈ 0.02603 mm.
- Process capability: Cp = T_WC / (6 · σ_RSS) = 0.13 / (6 × 0.02603) ≈ 0.832. The assembly is statistically capable (σ_level = Cp × 3 ≈ 2.5σ); for safety-critical work upgrade individual tolerances until Cp ≥ 1.33.
Illustrative example — verify against your actual nominal dimensions, tolerances, and assembly direction. The minimum WC gap of 2.870 mm guarantees no interference for any combination of parts produced within tolerance.
Frequently asked questions
Which standard does this tolerance stack-up calculator use?
The worst-case and RSS stack-up methods follow ISO 14253-1:2017 (verification by measurement — decision rules for proving conformance or non-conformance to specifications). The shaft/bore fit analysis and standard deviation designations follow ISO 286-1/-2:2010 (limits and fits for cylindrical parts). The process capability index uses the one-sided formula Cp = T_WC / (6 · σ_RSS), where T_WC = Σ t_i is the worst-case half-band and σ_RSS = √Σ(t_i / 3)².
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 Cp ≥ 1.33 check to confirm the process can meet the spec.
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 tolerance grades (IT01–IT18). You enter the bore and shaft deviations directly, or pick a standard fit pair such as H7/h6 (close clearance), H7/k6 (transition), or H7/p6 (interference). The calculator returns the maximum and minimum clearance, classifies the fit type, and — for interference fits — estimates the Lamé contact pressure for a steel solid shaft in a like-material hub.
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 part of a 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 Analysis (DIN 743)Verify 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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