Unilateral & Asymmetric Tolerances in Stack-Ups
Not every tolerance is ±0.1. Real drawings have unilateral tolerances (+0.05/−0), asymmetric tolerances (+0.10/−0.05), and limit dimensions (25.00–25.15). This guide shows how to handle these non-symmetric cases correctly in your tolerance stack-up.
Tolerance Types
Engineering drawings express tolerances in several ways:
| Type | Example | Meaning |
|---|---|---|
| Bilateral (symmetric) | 25.00 ±0.10 | 24.90 to 25.10 |
| Unilateral (positive) | 25.00 +0.10/−0 | 25.00 to 25.10 |
| Unilateral (negative) | 25.00 +0/−0.10 | 24.90 to 25.00 |
| Asymmetric | 25.00 +0.15/−0.05 | 24.95 to 25.15 |
| Limit dimensions | 25.00–25.20 | 25.00 to 25.20 |
Why Asymmetric Tolerances Exist
Designers use non-symmetric tolerances for good reasons:
- Process bias — A machining process consistently cuts slightly oversize, so the tolerance band is shifted
- Functional need — Clearance must exist but can be larger (hole: +tolerance only)
- Material removal — Stock can be removed but not added (grinding: −tolerance only)
- Fit control — ISO fits inherently have asymmetric tolerance zones
- Assembly direction — A part can be too long (interference) but not too short (clearance OK)
Converting to Mean and Half-Width
For stack-up calculations, it's useful to convert any tolerance to a consistent form: mean dimension ± half-width.
Mean = Nominal + (Upper + Lower) / 2
Half-width = (Upper − Lower) / 2
Where Upper and Lower are the deviation values (positive for + tolerance, negative for − tolerance).
Example Conversions
| Original | Upper | Lower | Mean | Half-Width |
|---|---|---|---|---|
| 25.00 ±0.10 | +0.10 | −0.10 | 25.00 | ±0.10 |
| 25.00 +0.10/−0 | +0.10 | 0 | 25.05 | ±0.05 |
| 25.00 +0/−0.10 | 0 | −0.10 | 24.95 | ±0.05 |
| 25.00 +0.15/−0.05 | +0.15 | −0.05 | 25.05 | ±0.10 |
| 25.00–25.20 | +0.20 | 0 | 25.10 | ±0.10 |
Why Convert?
The mean/half-width form lets you use standard RSS formulas. The mean becomes your nominal for the stack, and the half-width is your symmetric tolerance contribution. WC still uses the actual limits.
Effect on Worst-Case Analysis
Worst-case analysis uses the actual limits, not the mean. For each dimension:
- Maximum gap: Use upper limit for + contributors, lower limit for − contributors
- Minimum gap: Use lower limit for + contributors, upper limit for − contributors
The asymmetry of individual tolerances directly affects where the gap limits land.
WC Example
Consider a two-dimension stack: Gap = A − B
| Dimension | Specification | Min | Max | Sign |
|---|---|---|---|---|
| A (housing) | 50.00 +0.15/−0.05 | 49.95 | 50.15 | + |
| B (insert) | 49.50 +0/−0.10 | 49.40 | 49.50 | − |
Max Gap = Amax − Bmin = 50.15 − 49.40 = 0.75 mm
Min Gap = Amin − Bmax = 49.95 − 49.50 = 0.45 mm
Note how the asymmetric tolerances shift the gap range. If both had been symmetric ±0.10, the range would have been different.
Effect on RSS Analysis
RSS assumes a normal distribution centered on the mean. For asymmetric tolerances:
- Convert each dimension to mean ± half-width
- Use the means to calculate nominal gap
- Use the half-widths in the RSS formula
RSS Example (Same Dimensions)
| Dimension | Original | Mean | Half-Width |
|---|---|---|---|
| A (housing) | 50.00 +0.15/−0.05 | 50.05 | ±0.10 |
| B (insert) | 49.50 +0/−0.10 | 49.45 | ±0.05 |
Nominal Gap = 50.05 − 49.45 = 0.60 mm
RSS tolerance = √(0.10² + 0.05²) = √0.0125 = ±0.112 mm
RSS Range: 0.49 – 0.71 mm
RSS Assumption Warning
RSS assumes the process is centered on the mean. If your actual process is biased (e.g., consistently at the upper limit), RSS predictions may be optimistic. For critical applications, verify process centering with data.
Common Drawing Callouts
Here's how asymmetric tolerances typically appear on drawings:
| Callout | Interpretation | Common Use |
|---|---|---|
| Ø10.00 +0.02/−0 | Hole can only be larger | Reamed holes (clearance fits) |
| Ø10.00 +0/−0.02 | Shaft can only be smaller | Ground shafts (clearance fits) |
| 25.00 +0.05/−0.15 | Biased toward undersize | Machining before finishing |
| Ø10H7 | ISO fit (inherently asymmetric) | Standard hole-basis fits |
| 25.0 MIN | No upper limit (or title block max) | Minimum material requirements |
| 25.0 MAX | No lower limit (or title block min) | Maximum material requirements |
Worked Example: Piston in Bore
A piston slides in a cylinder bore. The clearance must be 0.02–0.08 mm for proper oil film and sealing.
Given Dimensions
| Feature | Specification | Min | Max |
|---|---|---|---|
| Bore diameter | Ø80.00 +0.03/−0 | 80.00 | 80.03 |
| Piston diameter | Ø79.95 +0/−0.02 | 79.93 | 79.95 |
Worst-Case Clearance
Max Clearance = Boremax − Pistonmin = 80.03 − 79.93 = 0.10 mm
Min Clearance = Boremin − Pistonmax = 80.00 − 79.95 = 0.05 mm
WC range is 0.05–0.10 mm. This exceeds the 0.08 mm max spec at worst case.
RSS Clearance
Converting to mean ± half-width:
- Bore: mean = 80.015, half-width = ±0.015
- Piston: mean = 79.94, half-width = ±0.01
Nominal Clearance = 80.015 − 79.94 = 0.075 mm
RSS tolerance = √(0.015² + 0.01²) = ±0.018 mm
RSS Range: 0.057 – 0.093 mm
RSS predicts most assemblies will have 0.057–0.093 mm clearance—closer to spec but still exceeding 0.08 mm at the upper end. Consider tightening the bore tolerance to meet spec.
Common Pitfalls
- Using nominal instead of mean — For asymmetric tolerances, nominal ≠ mean. RSS requires the mean.
- Treating unilateral as symmetric — A +0.10/−0 tolerance is NOT ±0.05. The total width is 0.10, but the mean is shifted.
- Forgetting sign effects — The direction of asymmetry matters. A +0.10/−0 on a positive contributor helps maximum gap, hurts minimum gap.
- Mixing formats — Be consistent. Either convert everything to mean ± half-width, or track upper/lower limits for each dimension.
- Ignoring process centering — RSS assumes centered distributions. Unilateral tolerances often exist because the process isn't centered.
Key Takeaways
- Asymmetric tolerances shift the mean — Account for this in RSS calculations
- WC uses actual limits — No conversion needed, use min and max directly
- Convert to mean ± half-width for RSS — This centers the distribution correctly
- The total tolerance band is Upper − Lower — Not 2× the nominal tolerance
- Verify process centering — RSS assumes centered; unilateral tolerances may indicate biased processes