Tolerance Stack-Up Calculator (Worst Case and RSS)
Nominal gap and the smallest and largest gap of an assembly of up to five parts, by worst-case arithmetic and by the statistical root-sum-square method.
Results
What this tool does
Enter each dimension in the chain with a plus sign if it opens the gap and a minus sign if it closes it, together with its ± tolerance. The worst-case stack adds the tolerances arithmetically, so it holds for every assembly ever made but is expensive; the root-sum-square method treats each tolerance as about ±3σ of an independent, centred, normal variation and gives ±√Σt², so four equal tolerances stack to twice one, not four times. Bender's factor of 1.5 widens the statistical figure to allow for drift. Use worst case for safety-critical or low-volume work.
Formula
nominal gap = Σ dimensions · worst case = ± Σ|t| · statistical (RSS) = ± √(Σ t²) · Bender = ± 1.5 √(Σ t²)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
a1 | Dimension 1 (+ adds to the gap, − closes it) | mm |
t1 | Tolerance ± on dimension 1 | mm |
a2 | Dimension 2 (+ adds to the gap, − closes it) | mm |
t2 | Tolerance ± on dimension 2 | mm |
a3 | Dimension 3 (+ adds to the gap, − closes it) | mm |
t3 | Tolerance ± on dimension 3 | mm |
a4 | Dimension 4 (+ adds to the gap, − closes it) | mm |
t4 | Tolerance ± on dimension 4 | mm |
a5 | Dimension 5 (+ adds to the gap, − closes it) | mm |
t5 | Tolerance ± on dimension 5 | mm |
WC | Worst-case tolerance on the gap | mm |
NM | Nominal gap (sum of the dimensions) | mm |
RS | Statistical tolerance (root-sum-square) | mm |
BN | Bender's tolerance (1.5 × RSS) | mm |
GW | Smallest gap, worst case | mm |
GX | Largest gap, worst case | mm |
GR | Smallest gap, statistical | mm |
GS | Largest gap, statistical | mm |
Worked example
- Dimension 1 (+ adds to the gap, − closes it)50 mm
- Tolerance ± on dimension 10.1 mm
- Dimension 2 (+ adds to the gap, − closes it)-20 mm
- Tolerance ± on dimension 20.05 mm
- Dimension 3 (+ adds to the gap, − closes it)-15 mm
- Tolerance ± on dimension 30.05 mm
- Dimension 4 (+ adds to the gap, − closes it)-10 mm
- Tolerance ± on dimension 40.03 mm
- Dimension 5 (+ adds to the gap, − closes it)0 mm
- Tolerance ± on dimension 50 mm
- Worst-case tolerance on the gap0.2300 mm
- Nominal gap (sum of the dimensions)5.0000 mm
- Statistical tolerance (root-sum-square)0.1261 mm
- Bender's tolerance (1.5 × RSS)0.1891 mm
- Smallest gap, worst case4.7700 mm
- Largest gap, worst case5.2300 mm
- Smallest gap, statistical4.8739 mm
- Largest gap, statistical5.1261 mm
Limitations
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.