Assembly tolerance stack-up: worst case and statistical methods

What a stack-up answers

A tolerance stack-up adds the variation of every feature that sits between two points of interest. The points might be the top of a terminal block and the rib on a cover, the edge of a busbar and the wall of a slot, or the centre of a mounting hole and the centre of the screw it must accept. The answer is a range, not a number.

Assemblies fail in the gap between drawings. Each part passes its own inspection, then the combination does not close, does not clear or does not sit flat. The stack-up is the calculation that finds that gap before hardware exists, and it is cheap compared with finding it on a pilot line.

Do the stack-up on the assembly drawing, not on a single component drawing. A component drawing cannot see the shift introduced by a fixture, a fastener or a moulding that shrinks differently in two directions.

Worst case: every tolerance at its limit

The worst case method assumes every contributor reaches its limit in the same direction at the same time. If five features each vary by plus or minus 0.10 mm, the total variation is plus or minus 0.50 mm. The arithmetic is a sum of absolute values, which makes it easy to explain to a supplier and easy to defend in a review.

The method describes a part that is extremely unlikely to exist, because it requires every process to drift outward together. That is its strength and its cost. It gives a guaranteed limit, and on a tight assembly it can force tolerances that are expensive to hold and unnecessary in practice.

Worst case is the right choice when the consequence of a single failure is severe: a live part too close to a cover, a sealing face that must compress, a latch that must engage every time. In those cases the low probability of the extreme is not a comfort.

Present the result as a table with one row per contributor, the nominal, the tolerance and the direction. A reviewer can then see which row to attack. A single number with no breakdown cannot be reviewed, and it usually gets accepted without one.

Root sum square: the statistical view

The root sum square method squares each tolerance, adds the squares and takes the square root. A stack of five contributions of 0.10 mm gives 0.22 mm rather than 0.50 mm. The narrower answer comes from an assumption: the contributors vary independently and sit around the nominal, so the extremes rarely coincide.

The assumption is reasonable for a mature process with a centred distribution. It is not reasonable for a process that is skewed, for a moulding that always shrinks in one direction, or for a stack with only two contributors, where the statistical argument has little to work with.

Root sum square also says nothing about the share of parts outside the limit. It narrows the predicted range, so a small percentage of assemblies may still fall outside. Where that share matters, the calculation needs a capability figure behind it, not just the tolerance values.

Where the data exists, replace the assumed distribution with the measured one. A supplier that has run the part for a year has real spread data, and using it turns the stack from an estimate into a prediction.

Choosing a method

Most programmes need both answers. Run the worst case first to see whether the design closes at all, then run the statistical method to see how much margin the process really has. If worst case passes, the statistical result is a comfort. If worst case fails and the statistical result passes, the design depends on process control, and that dependency should be a written decision rather than an accident.

The contributors people leave out

Most stacks are wrong because they are incomplete, not because the arithmetic is wrong. The omissions tend to be the same ones.

• Plating and coating: a 10 micrometre layer on two mating faces changes an interference fit, and thick coatings bridge radii.
• Moulding shrink direction: a housing shrinks differently along the flow and across it, so one tolerance does not describe both axes.
• Fastener true position: a clearance hole does not guarantee the screw lands in the insert if the insert has its own position tolerance.
• Fixture and assembly shift: a part located by hand in a jig moves within the clearance of the jig, and that movement belongs in the stack.
• Thermal expansion: a stack that closes at 20 degrees Celsius can interfere at 80 degrees if the materials differ.
• Measurement uncertainty: the gauge that accepts the part has its own error, and a marginal part is accepted on both sides of the limit.

A worked stack on a three part assembly

The example below is illustrative, but the structure is typical. A terminal block sits in a pocket in a moulded housing, and a rib on the cover passes above it. The design needs a minimum clearance of 0.80 mm between the top of the block and the tip of the rib, with a nominal of 1.20 mm.

Five contributors sit in the chain: the pocket depth in the housing at plus or minus 0.15 mm, the height of the terminal block at plus or minus 0.10 mm, the height of the cover rib at plus or minus 0.12 mm, the position of the insert that sets the block height at plus or minus 0.08 mm, and the shift at the cover joint at plus or minus 0.05 mm.

Worst case gives plus or minus 0.50 mm, so the clearance runs from 0.70 mm to 1.70 mm. The minimum requirement is 0.80 mm, so the design fails at the low end. Root sum square gives plus or minus 0.24 mm, so the clearance runs from 0.96 mm to 1.44 mm, and the design passes.

The decision is now explicit. Either accept that the minimum clearance depends on the processes staying centred, or buy margin. The cheapest margin is usually to raise the nominal by 0.15 mm in the cover rib, which costs nothing in tooling if it happens before the steel is cut and costs a tool change if it does not.

Turning the result into a drawing call

A stack-up that stays in a spreadsheet changes nothing. The result has to land on the drawing as a tolerance on a specific feature, with a datum that a supplier can measure from.

Tighten the largest contributor, not all of them. In the example, the pocket depth and the rib height carry more than half the variation. Reducing both by 0.03 mm does more than shaving 0.01 mm from four smaller items, and it is easier to inspect.

Then check that the drawing says what the calculation assumed. If the stack used a flatness call on the cover interface, the drawing must carry it. If the stack assumed the parts are measured at 20 degrees Celsius, the drawing should say so. A tolerance that exists only in the model is not a tolerance.