How tolerance stack calculations are performed

1.0 Introduction

The purpose of this literature report get a good grasp on how tolerance stack calculations are performed. The main article to be reviewed is ‘Tolerance Stack Analysis Methods’ by Fritz Scholz for the Research and Technology department of Boeing Information & Support Services located in Seattle, United States, released in 1995. The methods explained in this report are still maintained by most manufacturing companies nowadays.
Technology has however developed a solution in the meantime. Several software developers such as 3DS and Siemens made 3D tolerance analysis software available to the industry. Because this software could eliminate a lot of waste on excessive (re-)design, modern companies are evaluating its potential in order to get rid of their current ways of calculating tolerances stacks. In order to know if the outcome of the software is any good, calculation knowledge of tolerance stacks has to be reviewed for understanding how the software operates.

1.1 Background
The technology has undergone major changes over the centuries to meet the changing requirement of the society. During World War II, the United States manufactured and shipped spare parts overseas for the war effort. Many of these parts were made to specifications but would not assemble. The military recognized that producing parts that do not properly fit or function is a serious problem since lives depend on equipment that functions properly. After the war, a committee representing government, industry, and education spent considerable time and effort investigating this defective parts problem; this group needed to find a way to insure that parts would properly fit and function every time. The result was the development of GDT.
The report of Fritz Scholz), Tolerance Stack Analysis Methods, refers to various books and papers dating back to 1951, which on their time referred to even older references, for example Gramez (1925). None of the known sources dating before 1951 were obtained by the author, thereby they were not directly implemented in his report. The report also states that several companies protected their research (e.g. IBM), whereas other made them widely available (e.g. Motorola).
A lot of work has been done in the field of conventional tolerancing. Conventional tolerancing methods do a good job for dimensioning and tolerancing size features and are still used in good capacity today, but conventional tolerancing do not cater precisely for form, profile, runout, location and orientation features. Geometric Dimensioning and Tolerancing is used extensively for location, profile, runout, form and orientation features.
1.2 Relevance

The tolerance stacking problem arises in the context of assemblies from interchangeable parts because of the inability to produce or join parts exactly according to nominal. That is why tolerance stack calculations are relevant for any company that deals with assembling parts. If these calculations are not executed well, companies cannot guarantee if the parts will fit together in assembly. And therefore they could not guarantee the quality of their products.

1.2.1 Relevance to Inalfa
The use of tolerance stack calculations is of high importance to Inalfa Roof Systems. Because her clients (Car manufacturers) maintain high quality standards to assure good products and small rejection, the same quality standards are also required from their suppliers.
The standard of ‘good quality’ at Inalfa is set on a Cpk (Centered Process Capability) of 1,33, which is corresponding to a 4σ level. This means 63 out of a million products is allowed to be rejected.
Currently Inalfa enters the separate part tolerances in an Excel file, which then adds up the tolerance stacks to an assembly tolerance. These tolerance sheets are referred to as ‘TSS’ (Tolerance Stack Sheets). Basically, you only have to fill in the part tolerances and the excel file will provide information regarding the tolerance which contributes the most to the tolerance, and the capability of product.
These results are often not as accurate as a simulation could make possible, and they take a lot of time. Therefore Inalfa decided it wants to replace the excel sheets with 3D tolerance analysis software. To confirm the results the software provides it will come in handy to understand how tolerance stacks are calculated.
2.0 Dimensioning & Tolerancing

2.1 Dimensioning
A dimension is a numerical value expressed in appropriate units of measure and used to define the size, location, orientation, form, or other geometric characteristics of a part.
A tolerance is the total amount that features of the part are permitted to vary from the specified dimension. The tolerance is the difference between the maximum and minimum limits. When adding tolerances on dimensions there are several ways to display them:
– A bilateral tolerance is one that allows the dimension to vary in both the plus and minus directions. An equal bilateral tolerance is where the allowable variation from the nominal value is the same in both directions
– A unilateral tolerance is where the allowable variation from the target value is all in one direction and zero in the other.
– An unequal bilateral tolerance is where the allowable variation is from the target value, and the variation is not the same in both directions.
2.1.1 Fundamental Dimensioning Rules
The fundamental dimensioning rules are a set of general rules for dimensioning and interpreting drawings. The ten rules that apply to drawings are paraphrased in the list below:
1. Each dimension shall have a tolerance, except those dimensions specifically identified as reference, maximum, minimum or stock size
2. Dimensioning and tolerancing shall be complete so there is full definition of each part feature
3. Dimensions shall be selected and arranged to suit the function and mating relationship of a part and shall not be subject to more than one interpretation.
4. The drawing should define a part without specifying manufacturing methods.
5. A 90° angle applies where centerlines and lines depicting features are shown on a drawing at right angles, and no dimension is shown.
6. A 90° basic angle applies where centerlines of features in a pattern – or surfaces shown at right angles on a drawing — are located and defined by basic dimensions, and no angle is specified.
7. Unless otherwise specified, all dimensions are applicable at 20°C (68° F).
8. All dimensions and tolerances apply in the free-state condition. This principle does not apply to non-rigid parts.
9. Unless otherwise specified, all geometric tolerances apply to the full depth, length, and width of the feature.
10. Dimensions and tolerances apply only at the drawing level where they are specified. A dimension specified on a detail drawing is not mandatory for that feature on the assembly drawing.

2.2 Tolerancing

The tolerance system is the universal language for precision and quality spoken between the designer and the operator. If the system is used properly, it has the potential to decrease design costs effectively.
Tolerances are mostly read from 2D drawings. This should be shown in a universal way, so that everybody involved in the product understands the annotations. There are two standards to this norm. ISO standards are the international standards, and an American standard; the ANSI standards. These standards may vary slightly.
Commonly a 3D drawing is exported to a 2D drawing. Then, when all the correct views are added to the 2D drawing, the tolerances are added. Tolerances can be displayed in different kind of ways. Sometimes a flag note is added to redirect the reader to more information about the component, but in most cases a leader is used in extend to a feature control frame containing information about the component.

2.2.1 Feature Control Frame
The frame contains the following information:

Figure 1, Feature Control Frame
Geometric Characteristic symbol
These symbols illustrate the kind of tolerance, for all symbols used in geometric dimensioning and tolerancing see appendix 7.1. These symbols differ slightly to the ANSI standards.
Optional diameter symbol
The diameter symbol is always in front of the tolerance value. It can be used only for circular components like pins and holes.
Tolerance value
The tolerance value is the difference between the maximum allowed tolerance and the nominal value. In tolerance frames this value is always bilateral.

Material conditions of tolerance
If you want to ensure that two parts never interfere, or limit the amount of interference between the parts when they are at their worst tolerances, ‘Maximum Material Condition’ (MMC), or ‘Least Material Condition’ (LMC) can be called out.
If it is a hole or internal feature: MMC = smallest hole size, LMC = largest hole size
If it is a pin or external feature: MMC = largest size of the pin, LMC = smallest size of the pin
Material conditions are depicted with a circled M or L. If a circled S (or no symbol) is shown it means the tolerance is regardless of feature size (RFS). This term indicates a geometric tolerance that applies at any increment of size of the feature within its size tolerance. Another way to visualize RFS is that the geometric tolerance applies at whatever size the part is produces. There is no symbol necessary for RFS because it is the default condition for all geometric tolerances.

Primary, secondary, and tertiary datum references
When specifying a non-datum related control, the feature control frame will have two compartments. When specifying a datum related control, the feature control frame may have up to five compartments.
A datum is a virtual ideal plane, line, point or axis. A datum feature is a physical feature of a part identified by a datum feature symbol and corresponding datum feature triangle, e.g.:
Tolerances are referred to datum references which indicate measurements that should be made with respect to corresponding datum feature. A datum feature often is the surface where the component makes contact with the jig or fixture.
1. Primary Datum Plane: The primary datum is selected to provide functional relationships, standardizations and repeatability between surfaces. A standardization of size is desired in the manufacturing of a part. Consideration of how parts are orientated to each other is very important. The chosen primary datum must insure precise measurements.
2. Secondary Datum Plane: Secondary data are produced perpendicular to the primary datum so measurements can be referenced from them.
3. Tertiary Datum Plane: Tertiary data are always perpendicular to both the primary and secondary data ensuring a fixed position from three related parts.

3.0 Problem Formulation

To view the problem correctly, section views containing all the parts necessary for the stack-up calculation are made. A simple example is made in figure 1.

Figure 1 displays a section cut of several cogwheels stacked on a shared axle, covered by a fixed case.
3.0.1 Tolerance Gap
Because the cogwheels need to fit in the case, the available space (Lâ,) needs to be larger than the separate parts combined (Lâ,, + Lâ,ƒ + Lâ,” + Lâ,… +Lâ,†). In other words, it is desired to have a gap greater than zero (G > 0). Because the actual lengths Láµ¢ (i as in number of part) may differ from the nominal lengths λᵢ, there may well be significant problems in satisfying G > 0.
In some occasions it may be wishful to get G as close to 0 as possible. Or even have a negative gap if the goal is to get the parts ‘stuck’ in the available room.

3.0.2 Direction of Tolerance

It is very important to distinguish the negative from the positive tolerances, as one mistake directly has a big influence on the gap. A tolerance chain graph makes it easier to verify if a dimension negative or positive. Figure 2 displays a tolerance chain graph. Setting up a tolerance chain graph helps you overview all of the tolerances and their directions.

3.1 Equations

To calculate the gap (G) given in figure 1 & 2 the following equation is used:

To control the variation in dimensions, we limit part dimensions through tolerances. Such tolerances (Tᵢ) represent an “upper limit” on the absolute difference between actual and nominal values of the part dimensions. This is noted as:

Where the nominal value of the part’s dimensions is indicated by ‘λᵢ’, the nominal value of the gap (G) is indicated by ‘γ’. γ can be found when replacing the Lᵢ values in equation 1 by λᵢ:

If the objective is to achieve a gap that is positive, but in limits, a tolerance can be given to the gap. The tolerance (G — γ) given to the gap can be expressed as follows in terms of: .

4.0 Quicky GDT model

In the previous example, all tolerances given in the drawing were included in the calculation. This is not always the case. In more detailed drawings one does not always use all tolerances in the total assembly calculation. It is imported to ascertain which parts are attached to each other. The ‘Quicky GDT model’ was designed to make a better understanding for distinguishing the tolerances. As shown in figure 3 and 4.

Figure 3 shows 2 blocks stacked on top of each other, the arrows show the place where the tolerance is related to. When defining the tolerance path, tolerance 2a and 2b should not be taken in account, as part 1 is mounted in the middle of part 2 and therefore bridges those tolerances.

When following the path from top to bottom, stacking up all the relevant tolerances apart from the dimensions, the total assembly will have a dimension of 25.0 mm with a tolerance zone of ± 0.66 mm.
With this information we can conclude that:
Maximum Material Condition (MMC) = 25.66
Least Material Condition (LMC) = 24.33.
5.0 Formulas

In this section various formulas are describes to calculate the tolerance of the complete assembly. This means we combine the part tolerances (Tᵢ) into an assembly tolerance (Tassy¬).

5.1 Arithmetic or Worst Case Tolerance Stacking

When tolerances stack up, and the situation would be that all of the components tolerances were produced on the verge of their limits, the product is in a ‘Worst Case’ or ‘Arithmetic’ situation. The chance that this would occur is minimal; especially when there are many different components (as the number of components in the assembly increases then the chances of all the individual tolerances occurring at their worst case limits reduce). Yet it is necessary to calculate the worst case situation, to help define the limits of the tolerances and achieve as little as possible failure of products in practice.
The way to calculate the worst case is simply adding up the largest tolerances, just as we’ve seen in the Quicky GDT model. In formula this would be:

Figure 9, the Arithmetic tolerance stack formula
The worst case scenario is used for determining the Maximum Material Condition (MMC) and Least Material Condition (LMC).

5.2 RSS Method or Statistical Tolerancing
Another method used to calculate tolerance stacks is the RSS, or ‘Root Sum Square Approach’. The RSS method assumes that detail variations are random and independent. It takes random differences between part lengths and their nominal, squares these differences, sums the differences and then takes the root out of it.
The Root Sum Square Approach is applicable when the number of constituent dimensions in assembly is sufficiently large; the volume of production is very high and finite rejection of the product assembly is acceptable.

Figure 10: The Root Sum Square Formula
What we’re going to end up with is a distribution. The midpoint of the distribution will be the mean, and the distribution will between plus or minus 3σ. This distribution will reflect the expectation of the amount of product that will be produced within limits.

5.3 RSS method with Inflation Factors
In practice the RSS or Statistical tolerancing method seems to be a bit too optimistic. More products are out of specification than promised. This means that actual assembly stack variations are wider than indicated. Therefore a safety factor often gets build in in the formula, to prevent bad product quality.
This is because the formula doesn’t include several variations.
– Independence (no relations between parts)
– Temperature of the part/assembly
– Tool wear
– Possibility of bad inspection
– Sensitivity or importance of the stack.
– 3σi =Ti is not fulfilled
– Process is not centered
It is difficult to cope with these variations in any statistical way, which is why these variations are often compensated by a correction factor in front of the formula to solve the problem. If e.g. 3σi =Ti is not fulfilled, because process owners sometimes will respond with a ±Ti value which corresponds to a±2σi
Range, the formula needs a correction of 1.5. This correction is called ‘Benderizing’, named after professor Bender (1962).

Figuur 11: Benderized RSS Method
6.0 Distribution

6.1 Normal Distribution
There can be differences in the distribution over the tolerances intervals. The most common distribution is the normal distribution. The normal distribution shows that if we take random dimensions within the tolerance zone for all of the components, the most common outcome is that the tolerance of the assembled products will be centered at the mean.
Adding an inflation factor to the formula will alter the curve, figure displays derivations of the normal distribution:

6.2 Mean Shifts
However a centered mean is the desired outcome, it may not always be the case that it is. The mean can be off-centered for many reasons. The most common reason is that a supplier can easily produce products within the specified limits, and therefore choses to shift the mean towards the LMC to save costs on material while still producing within limits.
While maybe it wouldn’t seem to matter for one product to shift the mean while it still is between its limits, it can however influence the mean of the total assembly. This may cause more failures in the finished products, as the graph spreads out more to one side and crosses the limit.
Examples of mean shifts are given in figure16
6.2.1 Mean Shift Correction

Mean shifts are a systematic component of detail part deviation from nominal, and therefore they can be described in formula. In addition to the shift caused by supplier, a tool that is being used for assembling over and over wears out over time. This wear can be the cause of a mean shift. In most cases there is no way to prevent the tool from wearing, we can however, take this kind of variation into account in the assembly formula. A correction factor won’t do the trick this time, as it could correct too often it would possibly add more variability to the process. The right thing to do here is to allow some kind of mean shift. The process variation still has to be within limits.

6.2.2 Mean Shift Correction Formula

If the choice is made to include mean shifts in the formula, they have to be added arithmetically. The ultimate formula to calculate tolerance stacks would then be the arithmetic stack of mean shifts + part to part variability via the RSS method. This brings us to the following formula:

If η=0 then there is no mean shift and the formula will give the same results as Tstat,assy.
When η=1 it means that the mean is shifted all the way to the tolerance limits. In this case the formula will give the same results as Tarith,assy.  
7.0 Conclusion

There are different ways to calculate tolerance stacks. There is not one specific best way to calculate tolerance stacks, it all depends on what assumptions one is willing to make. The arithmetical or worst case calculation is probably the safest way to go if tolerances are used for protection purpose. The downside of this calculation is that one will end up with fairly wide assembly tolerance limits. If many data is available and the goal is to predict how many parts/products will be rejected in the assembly process, the RSS method is recommended. If the results of the RSS method seem to be too optimistic, the formula needs to be corrected with an inflation factor.
Constant comparison between the formula graphs and actual results is necessary in order to see if the results correspond, because the RSS calculations can never precisely enough forecast the outcome. If the mean seems to be off-centered, the mean shift correction formula is the best way to get the right calculations.

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