Tolerance stack-up for sheet metal: worst case vs RSS

Two brackets from the same drawing, one measuring 149.6 mm and one 150.5 mm, meet in the middle of an assembly that has 0.6 mm of clearance. Both are inside their individual tolerances. The assembly does not close, and the argument that follows is about whether the drawing or the supplier is at fault, when the real fault is that nobody added the six tolerances up before releasing the drawing.

Tolerance stack-up for sheet metal is arithmetic that most drawings never show. The two methods, worst case and root sum square, answer different questions. Worst case asks what happens if every part is at its limit in the worst direction. RSS asks what happens across a production run when errors are random and mostly cancel.

The gap between them is large enough to change a design decision. On a six-link stack each at ±0.3 mm, worst case is 1.84 mm and RSS is 0.99 mm. Understanding why, and when RSS is not allowed to be used, is the difference between a tolerance you can hold and one you have to negotiate.

What is the difference between worst case and RSS?

Worst case adds the tolerances linearly. RSS squares each tolerance, adds the squares and takes the square root. For links t1 to tn:

Worst case: T = t1 + t2 + ... + tn

RSS: T = square root of (t1² + t2² + ... + tn²)

Six links at ±0.3 mm give:

Worst case = 6 × 0.3 = 1.80 mm

RSS = 0.3 × square root of 6 = 0.3 × 2.449 = 0.73 mm

The ratio between the two methods is always the square root of the number of links, when the links are equal. That single relationship is the decision rule: with four links the ratio is 2.0, with six it is 2.45, with ten it is 3.16. Below four links, RSS buys less than a factor of two and the assumption load is not worth it.

RSS works because independent random errors rarely all sit at their extreme in the same direction. If each link is normally distributed with its tolerance equal to three standard deviations, the assembly result is also normal, with a standard deviation equal to the root sum square of the link standard deviations. The ±3σ assembly limit is then the RSS value, and 99.73% of assemblies fall inside it.

That last sentence carries the conditions, and they are strict:

1. The links are independent. A shared fixture, a shared setup or a shared material lot breaks this.
2. Each link is centred on its nominal. A process that runs at one side of the tolerance is a bias, not a random error.
3. Each stated tolerance really is a 3σ value for the process. A tolerance that is a drawing guess is not.

A six-link stack worked through

Take a welded bracket assembly: a laser-cut plate, two bends, a weld into a frame, and a powder coat. The critical dimension is the position of a hole in the frame relative to a hole in the plate, and the assembly needs it within ±1.0 mm. Each row contributes its full tolerance to the worst case and its square to the RSS sum, so the two columns only separate at the total.

The worst case total is 1.84 mm, which fails the ±1.0 mm requirement outright. The RSS total is 0.99 mm, which passes by 0.01 mm. A design that passes by 0.01 mm is not a design; it is a coin toss with a spreadsheet attached.

The table also shows where the money is. The bend angle contributes 0.87 mm, which is 47% of the worst case total and 76% of the RSS sum of squares before the root is taken. Reducing the other five tolerances by half would take worst case to 1.36 mm and RSS to 0.90 mm, which is barely a third of the gain from halving the bend angle. Reducing the bend angle tolerance from ±0.5° to ±0.25°, by forming on a controlled brake with a checked punch and die set and a first-off angle check, takes the same stack to:

Worst case = 0.20 + 0.20 + 0.44 + 0.30 + 0.20 + 0.07 = 1.41 mm

RSS = square root of (0.04 + 0.04 + 0.194 + 0.09 + 0.04 + 0.005) = 0.64 mm

That is a 35% cut in the RSS total from one decision, and it is the decision the shop can actually control. Chasing 0.05 mm off a laser tolerance costs a slower cut; halving bend angle error costs a check on the tooling.

Why bend angle dominates

Angular error converts to linear error by distance:

e = L × tan(Δθ)

At ±0.5°, tan is 0.00873, so:

• 50 mm from the bend: 0.44 mm
• 100 mm: 0.87 mm
• 300 mm: 2.62 mm
• 1,000 mm: 8.73 mm

A 2 m panel formed on a brake with 0.5° of angle variation has 17.5 mm of positional variation at the far end. No hole tolerance survives that, and no amount of laser accuracy helps, because the error is created after cutting.

Two consequences for the stack. First, if a feature is far from the bend that locates it, its position tolerance has to include the angle term, or the drawing has to dimension it from the bend line so the angle error does not accumulate. Second, on long parts the stack should be computed at the far end of the flange, not at a nominal 100 mm, because that is where the assembly closes.

When RSS is not allowed

RSS is a statistical statement, and it is wrong whenever its assumptions are wrong. Four cases where it must not be used as-is.

One link dominates. If one term is more than about 70% of the sum of squares, the distribution of the total is essentially the distribution of that link, and the square-root averaging across the others is misleading. Work on the dominant link first; the arithmetic will not save you.

The links are correlated. Parts located in the same fixture share the fixture error, and parts cut from the same nested sheet share the sheet position error. Correlated terms add linearly, not in quadrature. Remove them from the RSS, sum them, then add the result to the RSS total.

The processes are biased. Weld shrinkage pulls in a direction. Powder coating only adds thickness. A bending sequence that always forms the same flange first produces angle error with a consistent sign. Each of these is a mean shift, and the correct treatment is:

T = mean shift + k × RSS

where k is 3 for a 99.73% target. On the six-link example, if the weld pulls 0.3 mm one way and the coating adds 0.07 mm on the measured face, the shift is 0.37 mm, and the total becomes 0.37 + 0.99 = 1.36 mm. That is inside ±1.0 mm only if the shift is absorbed by clearance elsewhere.

The tolerance is not a process capability. A drawing tolerance of ±0.5 mm on a feature the process holds to ±0.2 mm is not a 3σ value; it is a limit. Using it in RSS overstates the stack. Using a capability figure the supplier has not measured understates it. Where the data does not exist, worst case is the honest method.

A rectangular distribution is the right model for a purchased or material-driven limit, such as a coating thickness range of 60 to 90 µm. Its standard deviation is the range divided by the square root of 12, which is 8.66 µm, so the 3σ value is 26 µm, not the 30 µm half-range. The difference is small here but large on a stack with several bought-in items.

A decision rule for the drawing

Use the following, in order:

1. Four links or fewer: worst case. The arithmetic is simple, the result is defensible, and the RSS saving does not justify the assumptions.
2. Five to eight independent, centred links with known capability: RSS plus a 20% margin. The margin covers process drift and the fact that the capability estimate is itself uncertain.
3. Any dominant term, any correlated terms, any bias: worst case, or a corrected RSS with the shift added linearly.
4. A safety-critical dimension, or a stack with more than ten links and a tight allowance: Monte Carlo. Ten thousand trials with the actual distributions takes minutes and shows the shape of the result, not just the total.

The practical finding from running these on sheet metal assemblies is consistent: the cutting tolerances are rarely the problem. The forming angle, the weld sequence and the coating are where the millimetres come from. A stack-up that spends its effort on the laser is optimising the smallest term.

What Ganesh runs

Stacks are checked against the route we would actually use, not a generic one. Cutting on the 12 kW fibre laser with the 40 ft × 16 ft bed holds ±0.2 mm position, with ±0.1 mm feature to feature in one setup. Forming on the 200 T, 4,450 mm, 5-axis brake with Delem control holds ±0.5° on angle, and the long bed matters because a part formed in one hit has one angle error rather than two. Welding runs on three CLOOS robotic cells and more than 30 manual stations, with fixtures built in the in-house tool room, which is also what makes the fixture term a real number instead of an assumption.

Coating is the last term and the one most often forgotten: seven-tank pretreatment, the 18 ft × 14 ft powder booth, and 60 to 90 µm of powder on each face, checked with a DFT gauge. Where a dimension closes on a coated face, the coating is in the stack whether the drawing says so or not.

First article inspection and PPAP are available for programmes that need the evidence, and the welded assembly tolerance guide covers how the weld terms are measured. The design and CAD hub collects the related datum and flat pattern rules, and the bending capability page lists the machine and control.

What to put on the drawing

• The critical assembly dimension, with its tolerance, marked as the stack result rather than left implicit.
• The datum scheme the stack was calculated from, so the inspector measures the same chain.
• Bend angle tolerance per bend, not one general note for the part.
• Position tolerances stated separately from size, because ISO 8015 treats size and position as independent unless the drawing invokes the envelope requirement.
• A general tolerance note to ISO 2768-1 for dimensions outside the chain, written so it cannot be read as covering the critical dimension.
• Weld symbols and sequence notes where shrinkage affects the critical dimension.
• A masking note for any datum or closing face that is coated.
• The process route assumed in the stack, since a different route gives a different answer.
• A note on which dimensions are reference, so inspection effort goes to the chain that matters.

When neither method fits

If the stack fails both ways, the answer is usually not a tighter tolerance. It is a design change: a slot instead of a hole, a shim, an adjustable joint, or a datum moved to the face that closes. A stack that needs ±0.1 mm on a welded frame is asking the process to do something it cannot, and no arithmetic fixes that.

We do not promise a stack result we have not run against the actual route, and we will say when the tolerance is outside what the machines and fixtures can hold. Send the drawing, the critical dimension and the annual quantity, and the stack can be checked before the first part is made. Request a quote with the revision, or read the datum strategy guide for how to choose the features the stack starts from.