TAHA Engineering Toolsby TAHA Knowledge Lab

Metrology & Quality

Tolerance Stack-Up Calculator (Worst Case)

Worst-case maximum and minimum gap of a 1-D dimension chain, with an RSS comparison.

Inputs

A U-shaped housing of inside length A containing two blocks B and C, leaving a gap at the right.BCA (+)B (−)C (−)gap
A one-dimensional chain: the gap equals the housing length A minus parts B and C. Dimensions that increase the gap count +, those that reduce it count −. Diagram not to scale.

Signed, e.g. +0.10. Empty = 0.

Signed, e.g. −0.05. Empty = 0.

Result

Minimum gap (worst case)
0.15
Maximum gap (worst case)
0.4
Nominal gap
0.2
Total tolerance (worst case)
0.25
Statistical range (RSS) — for comparison only
0.2 to 0.35 mm (mean 0.275, RSS tolerance 0.15)
Working
+ 50 (+0.1 / 0) mm - 30 (+0 / -0.05) mm - 19.8 (+0.05 / -0.05) mm Nominal = 0.2 mm; maximum = 0.4; minimum = 0.15 Worst-case tolerance = sum of tolerances = 0.25 mm (= max − min)

In the worst case the gap ranges from 0.15 mm to 0.4 mm (nominal 0.2 mm, total tolerance 0.25 mm).

Educational estimate — check the assumptions below.

About this calculator

When parts are assembled, their tolerances add up. A tolerance stack-up follows a chain of dimensions around a loop — for example a housing length minus the parts inside it — to find the size of a gap, clearance or overall length. This calculator handles a one-dimensional chain of up to eight dimensions.

The worst-case method assumes every part is at the limit that makes the gap largest, then at the limit that makes it smallest. If the minimum gap is still acceptable, every assembly will work — this is the safe method. The RSS (root-sum-square) range is shown only for comparison: it is narrower because it assumes the deviations are random and rarely all at their limits together.

Formula

Nominal gap = Σ (± Ni)

Maximum gap = Σ+(N + upper dev) − Σ−(N + lower dev)    Minimum gap = Σ+(N + lower dev) − Σ−(N + upper dev)

Worst-case tolerance = Σ Ti    RSS tolerance = √(Σ Ti²)

Σ+ = dimensions that increase the gap, Σ− = dimensions that reduce it, Ti = upper − lower deviation of each dimension. The RSS range is centred on the mean gap, using the middle of each tolerance.

Worked example

Gap = housing − part B − part C:

  • Housing 50.00 +0.10/0, part B 30.00 0/−0.05, part C 19.80 ±0.05 mm
  • Nominal gap = 50 − 30 − 19.8 = 0.20 mm
  • Maximum = 50.10 − 29.95 − 19.75 = 0.40 mm; minimum = 50.00 − 30.00 − 19.85 = 0.15 mm
  • Worst-case tolerance = 0.10 + 0.05 + 0.10 = 0.25 mm; RSS = √(0.1² + 0.05² + 0.1²) = 0.15 mm, centred on 0.275 mm → 0.20 to 0.35 mm

Assumptions and limitations

  • One-dimensional (linear) stack along a single direction. Angles, geometric tolerances (position, flatness, etc.), thermal expansion and deflection are not included.
  • The RSS result assumes independent, normally distributed deviations centred in their tolerance zones and a reasonably large number of parts. It is not a guarantee; use it only where your quality process supports it.
  • Each dimension in the chain should appear once. Draw a loop diagram first to decide which dimensions add (+) and which subtract (−).

References

  • Worst-case and root-sum-square tolerance analysis (e.g. ASME Y14.5 practice and standard engineering design texts such as Shigley’s Mechanical Engineering Design).