Tolerance stack-up analysis predicts how individual part variations combine to affect an assembly dimension such as a gap, clearance, preload, or interference fit. It matters because every manufactured part has allowable variation, and small deviations can add together to create a failed fit even when each part is within its own tolerance. Engineers use stack-up analysis to decide whether a design can be built reliably, inspected efficiently, and produced at a reasonable cost.
A stack-up begins by defining a dimension chain from one functional surface to another, then assigning each contributing dimension a sign based on whether it increases or decreases the final result. Worst-case analysis assumes all dimensions hit their most unfavorable limits at the same time, while statistical RSS analysis assumes independent random variations combine by root-sum-square. Tolerance allocation works in the reverse direction by dividing an allowable assembly variation among parts based on function, process capability, cost, and risk.
Key Facts
- Assembly result: R = d1 + d2 - d3 + d4, using signs from the dimension chain.
- Worst-case tolerance: T_wc = T1 + T2 + T3 + ... + Tn.
- Worst-case limits: R_min = R_nominal - T_wc and R_max = R_nominal + T_wc when T values are bilateral half-widths.
- Statistical RSS tolerance: T_rss = sqrt(T1^2 + T2^2 + T3^2 + ... + Tn^2).
- For independent normal dimensions, standard deviations combine as sigma_R = sqrt(sigma_1^2 + sigma_2^2 + ... + sigma_n^2).
- If a dimension has tolerance ±T and represents a 3 sigma process, then sigma = T / 3.
Vocabulary
- Tolerance stack-up
- A calculation that determines how part dimension variations combine to affect a final assembly dimension.
- Dimension chain
- The ordered path of dimensions that connects the two surfaces defining the functional assembly requirement.
- Worst-case analysis
- A method that assumes every part dimension is at its extreme limit in the direction that creates the largest or smallest assembly result.
- RSS analysis
- A statistical method that combines independent tolerances using the square root of the sum of their squares.
- Tolerance allocation
- The process of assigning allowable variation to individual part dimensions so the assembly requirement is met.
Common Mistakes to Avoid
- Adding all nominal dimensions without signs, because some dimensions increase the gap while others reduce it.
- Using RSS for dependent dimensions, because shared tooling, common datums, or process shifts can make variations correlated rather than independent.
- Treating total tolerance width as the plus-minus value, because a dimension listed as 20.00 ± 0.10 has a half-width of 0.10 and a full width of 0.20.
- Allocating equal tolerances to every part automatically, because critical surfaces, manufacturing processes, and cost sensitivity often require unequal tolerances.
Practice Questions
- 1 A gap is defined by G = A - B - C. If A = 50.00 ± 0.20 mm, B = 18.00 ± 0.10 mm, and C = 31.50 ± 0.15 mm, find the nominal gap and the worst-case minimum and maximum gap.
- 2 Four independent dimensions contribute to a critical length with bilateral tolerances ±0.05 mm, ±0.08 mm, ±0.04 mm, and ±0.06 mm. Calculate the worst-case tolerance and the RSS tolerance.
- 3 An assembly has a tight functional gap requirement, but one spacer is produced by a low-cost process with poor repeatability. Explain how tolerance allocation could be changed to improve yield without making every part more expensive.