With tolerance analysis you have several options for how you model and analyze the geometrical dimensioning and tolerancing (GD&T) values of your mechanical products and the stackups of their components. You can analyze in one dimension, two dimensions, or three dimensions — or a combination of these.

You can also to choose to work at a traditional tolerance analysis level, in which you focus on ensuring tolerance values that result in a good fit of components for assembly, or at a functional tolerance analysis level, in which you not only evaluate assembly fit but also the ability of the combination of all GD&T values to robustly meet a product’s performance requirements — such as for force levels or specific movements.

In this article we describe these 1D, 2D, and 3D stack-up modeling and analysis options, their differences, when to use them, and their tradeoffs for tolerance analysis. Here’s the contents:

- What is 1D stack-up tolerance analysis?
- What is 2D stack-up tolerance analysis?
- What is 3D stack-up tolerance analysis?

## WHAT IS 1D stack-up tolerance analysiS?

For a relatively simple mechanical design with components all stacked in a single direction, a 1D stack-up analysis can work well.

However, it is important to note that **a limitation of 1D analysis is that representing geometric aspects of a design such as perpendicularity, parallelism, or concentricity is very difficult or even not possible.** So if the assembled fit or functioning of your design are highly sensitive to geometric variations, you will want to go beyond 1D and move on to 2D or 3D stack-up analysis, which we describe below in this article.

As an example for 1D analysis, suppose you have a series of disks stacked together in the same direction within a housing enclosure. **If you are working with what is referred to as a worst-case stackup analysis,** then for each disk you’d set a plus/minus tolerance for its width and you’d also set a plus/minus tolerance for the housing opening that the stack fits into. You would then calculate the length of the stack twice — once when all the disks are set to the plus tolerance value, or when then they are at their thickest, and once when all the disks are at the minus tolerance value, or at their thinnest.

You’d determine if the stack at its longest and shortest possible lengths will still fit correctly into the housing at its largest and smallest possible opening lengths. In other words if the gap between the disk stack and the housing enclosure is within requirements. If everything fits, you’re good. If not, you will need to iterate on what tolerance values to use.

However, you might decide it is not practical or cost effective to set the tolerance values to be so small that you have the correct fit 100% of the time. Maybe you can live with a .1% failure rate for instance. In that case **you could use statistical techniques, such as RSS or Monte Carlo simulation, to estimate the probability that all the tolerances will add up in a way that the stack will not fit correctly into the housing**. You’ll iterate the tolerance values to achieve the .1% failure rate, which is a 99.9% success rate.

**Spreadsheets like Microsoft’s Excel are commonly used for this type of 1D stack-up analysis.**

## wHAT IS 2D tolerance analysiS?

### Limitations of spreadsheets

What if your product is more than, for example, a linear, 1D stack of disks and a housing enclosure? Instead, you’ve got moving cams, levers, and spring components that are all connected. That could be a product that is a part of an appliance, a car, an aircraft, or a medical device. The geometries quickly become complex. The geometric and dimensional tolerances can easily impact more than the fit of the combined components, they can affect the functionality of the product, such as the forces within and output by the product.

You could use a spreadsheet for analyzing tolerances. You might simplify your product, make assumptions, and then do 1D analyses from multiple perspectives. But that would quickly become difficult to create, understand, and maintain. It would take many individual sheets with lots of difficult to understand and hard to maintain formulas.

The formulas you create to capture geometries such as angles, parallelisms, and concentricities can be complicated and typically impractical in 1D. If you want to go beyond fit for assembly and represent functionality such as forces vs a mechanism’s motion, that is another layer of complicated formulas. **As the complexity of your assembly increases, the assumptions you make for 1D lead to modeling is far away from the reality of the product, which can easily lead to over simplification, error-prone results, and design decisions that lead to costly production and/or warranty problems.**

### Why 2D vs. 1D spreadsheets

**A better solution is to move from 1D to 2D using a dedicated tolerance analysis software product. **These programs are specifically designed for modeling and analyzing visually in two dimensions — these are not general-purpose Computer-Aided Design (CAD) programs that have tolerancing capabilities added on top of their main function.

These dedicated programs include geometric engine solvers that handle many more geometry types than what’s possible in spreadsheets. And they have automated worst-case and statistical analysis tools that perform calculations based upon user-defined sample sizes. The user feeds in geometries and tolerance values, selects analysis type(s) from menus, clicks buttons or fills out dialog boxes, and then immediately get outputs such as probability distributions of different types of failures for an entire mechanism.

### Functional Tolerance Analysis in 2D

Some of these software products go beyond telling you if your components will fit together as designed for assembly for a given set of tolerances. They also tell you if the assembled product will function as intended, such as delivering a specified range of motion or forces. **This is also known as functional tolerance analysis.** You can perform multiple analyses at once, looking for different failure modes and the probabilities of those modes. Once you identify the most important failure modes to address, you can focus on those areas of your design and iterate on the relevant tolerance values to fix it.

Enventive Concept is an example of this kind of a program built just for functional tolerance analysis. Working in 2D, design engineers visually model mechanisms and the functionality delivered. They apply multiple analysis techniques, including stack-ups, statistical, simulation, and animations to identify failure modes and estimate failure rates.

**With Concept, designers iterate not just on the values of component tolerances to reach desired results, they also can iterate on the dimensional and geometric values of the components and see results in real time.** For example by changing the length and angle of a lever arm the engineer might better avoid a costly failure mode.

By enabling rapid what-if analysis for any of the product’s GD&T parameter values, this kind of functional tolerance analysis tool enables engineers to work at a conceptual design level (thus the product’s name), which serves to speed up a design engineer’s decision making and increase their confidence in the results. Engineers can make more informed GD&T decisions across an entire design cycle — from before a detailed model is started in a 3D CAD system all the way to the end of CAD modeling and the creation of GD&T engineering drawings for manufacturing.

The result of functional tolerance analysis can be a greatly improved design process vs. traditional tolerance analysis, which is typically done near the very end of CAD modeling and is limited to validating that components fit together for assembling on the manufacturing floor.

## WHAT IS 3D tolerance analysis?

Suppose you’ve made the most important GD&T design decisions for your product, possibly with the use of 1D and/or 2D tolerance analysis methods. Those decisions have guided the creation of the 3D CAD model and GD&T drawings that are the blueprint for manufacturing each component and then assembling them into the final product. **How do you ensure that you have made tolerancing decisions that work for the reality of three dimensions?**

With a 3D tolerance analysis software product you can analyze the GD&T values that define each of your 3D CAD components separately and then analyze how they fit together. You feed 3D models into this kind of program, not just 2D drawings. This lets you take full advantage of the power to calculate stacks-ups, tolerances, superpositions, allowable combination of deviations, etc. across each feature on your product’s surfaces instead of being constrained to 1D planes as in spreadsheets.

3D tolerance analysis software typically involves much more detail and complexity versus 2D. It frequently requires skilled users with advanced training on the tool. It can be much more effort or even impractical to complete a functional tolerance analysis that involve iterations of GD&T parameters. 2D tools like Enventive’s Concept are better suited for this.

**More commonly 3D tolerance analysis works best as a validation tool to check for fit-related failure modes that would not be easily found with 1D or 2D analysis.** You usually do a 3D tolerance validation near the end of detailed CAD modeling. You can spot and resolve problems before building a physical prototype and going through the expense of testing. Because of the complexity of working in 3D, usually any changes made to GD&T values are relatively minor compared to those made at a conceptual 2D level. For this validation case, a 3D CAD model can be exported to a dedicated 3D tolerance analysis software program that does the checks for fit-related issues.

**In practice, we at Enventive find that our users will typically start with 2D functional tolerance analysis modeling and then use the results as a starting point for their 3D CAD models.** Frequently the 2D modeling will be completed before 3D CAD begins while other times 2D and 3D will be run concurrently, in parallel, during most of the 3D CAD design cycle.

Once a 3D CAD model is at or near its final version, a tolerance analysis can be run for validation with either a 2D or 3D tool. Again, for more complex mechanisms a 3D tolerancing validation may be the best choice.

For Enventive, **we find that the majority of the time our users get the tolerance analysis results they need in 2D without the need to go to 3D.** What we find is that while, of course, mechanism systems are 3D, their motions must be in 2D. So with mechanisms, 2D works very well to get good results and to get results quickly.

## Further reading

Cetim’s Tolerancing Handbook is a valuable reference for your GD&T analysis and drawings.