“The structure of the CrCoNi is the simplest you can imagine – it’s just grains,” Richie said. Co-author Professor Andrew Minor continues: “However, when you deform it, the structure becomes very complicated, and this shift helps explain its exceptional resistance to fracture.”

Mathematically tolerance zone is equal to the multiple of required process capability and total standard deviation for the assembly. Lower the value of process capability, narrow will be the tolerance zone and higher will be the rejection.

“The toughness of this material near liquid helium temperatures (20 °K, -424 °F) is as high as 500 megapascals square root meters. In the same units, the toughness of a piece of silicon is 1, the aluminum airframe in passenger airplanes is about 35, and the toughness of some of the best steels is around 100. So, 500, it’s a staggering number,” said Ritchie.

Can you provide a presentation or resources to guide in creating tol stack analysis stack diagrams, to include any assembly of 3 or more parts with a fixed fastener screw rectangular pattern? Simple disc stackup is clear but I am not clear how to handle more difficult aforementioned multipart ‘fixed fastener’ assy stack up.

The first step in doing tolerance stackup analysis is to create the dimensional chain. It is used to determine the direction of tolerance. In the above example, all dimensions are in a positive direction.

Similar to worst case tolerance analysis, the first step in RSS tolerance stackup analysis is to create the dimensional chain. It is used to determine the direction of tolerance. In the above example, all dimensions are in a positive direction.

How do you calculate if an element in the stack-up has unbalanced tolerance? For example X +0.5/-0.1 or X +1/-0? Do you shift the “nominal” and then balance the tolerance, or is there a more accurate way to calculate the final tolerance stackup?

Yes; It wont help machining supplier ; However this may help to casting supplier in case wall thickness calculation. If there is a defect on casting surface and it could lead to reduction in wall thickness. Due to RSS method machining tolerance will be reduced and in result increase in wall thickness.

Your spreadsheet is incorrect. For – 1.05 +0/-.01, the two values that are calculated are -1.06 and -1.05. The two values should be -1.05 and -1.04.

In this example, maximum and minimum disk stack-up height is calculated by adding nominal dimensions with upper and lower tolerance respectively.

Reference: Liu D, Yu Q, Kabra S, et al. Exceptional fracture toughness of CrCoNi-based medium- and high-entropy alloys at 20 Kelvin. Science. 2022;378(6623):978-983. doi: 10.1126/science.abp8070

According to the worst case tolerance stack up analysis disk stack height can vary from 65.5 mm to 68.5 mm. Whereas according to the RSS method disk height can vary from 66.2584 mm to 67.74162 mm.

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Root sum square (RSS) tolerance stack up analysis works on a statistical approach. It is assumed that most of the parts fall to the middle of the tolerance zone.

A team of researchers from the Lawrence Berkeley National Laboratory and Oak Ridge National Laboratory has identified the toughest material on Earth – a metal alloy of chromium, cobalt and nickel. Contrary to other materials, its strength increases as it gets colder.

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The researchers caution that although their progress in this field is exciting, new structural materials can take a long time to be used in real-world applications, as they need to be well-understood before they can replace well-tested and trusted existing materials.

The toughness of CrCoNi comes from the atomic structure of its crystalline lattice, which changes when force is applied to the alloy. A trio of atomic interactions occurs in sequence, followed by a rearrangement of the unit cells within the lattice, which enables CrCoNi to resist stress past the point where other materials would have fractured.

Sir, Can you help in calculate pin diameter and their distance tolerance for manufacturing tolerance for gauge design ? Assuming three holes are diameter 10+/- 0.1,diameter 15+/- 0.2 and diameter 25+/-.1 these holes distance is 120+/-1,100+/-0.5 and 80+/-1.5 respectively. Please share with formula to my Email

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For a good understanding of tolerance stack up. We suggest you first read this article on What are Limit, Fits, and Tolerance?

In RSS, the tolerace band is reduced. How it will help the supplier to produce parts. reduced tolerance band is the constraint for supplier, right?

Worst Case tolerance stackup analysis method utilizes simple arithmetic (addition and subtraction) operations to calculate optimized tolerances. In this method, all dimensions are assumed at the extreme limit. For low production volumes, worst case tolerance analysis is recommended.

I don’t understand the RSS scaling of the total tolerance band by 1/(2×3) to get the total standard deviation. If we assume the errors are uniform distributed – the standard deviation should = tolerance band/sqrt(12) = tolerance band/3.46. This is close to the factor of 3 used – but I don’t see where the additional factor of 2 comes in….

Thanks Edward for your feedback. This is a linear tolerance calculator, that simply works by adding the tolerance. Here -ve sign shows the tolerance direction. So ideally output shall be irrespective of sign. So i agree with you and we will improve on this point. But at the same time, i believe it has no impact on end results as this is calculated based on cumulative values. We recommend you to also check our online tolerance calculator where we tried to over come the limitations of excel: https://www.smlease.com/entries/product-design/online-tolerance-stackup-calculator/

Production of closely mating parts without tolerances is economically unfeasible. Tight tolerances can result in an increased rejection rate and manufacturing cost. Whereas loose tolerances can affect product function. Therefore it is always recommended to define optimized tolerances. Part tolerance optimization increases the part tolerance band. It also helps in reducing part manufacturing costs.

To sum up, Tolerance stackup analysis is a very important part of product design. It also helps in reducing part cost. Various tolerance stackup analysis methods are available.

Statistical tolerance analysis is based on the Bell curve and normal distribution. For example, when a thousand disks of the same size are manufactured. Disk thickness will vary from upper limit to lower limit. But all disks will not measure the same because their thickness will be distributed.

You can also use our worst-case and RSS method tolerance stack-up calculator for tolerance stack-up calculations. You can Download Tolerance stack-up calculation sheet here.

I do not have much knowledge of the sampling theory approach. But my understanding is we can use it since it is based on statics only.

Standard deviation is calculated by considering manufacturing process is 3σ capable. Mathematically standard deviation is equal to two times of process capability because the process is distributed on both sides. Therefore standard deviation for 3σ process is given by:

Statistical tolerance Analysis does not focus on the extreme dimensional limits. Because variation in manufactured part dimensions is not linear. Each dimension has a unique distribution based on the part manufacturing process, machines, and other parameters.

Toughness is the combination of strength and ductility (resistance to fracture) a material possesses. To measure toughness, a pristine specimen is pulled until it fractures, and the force needed to do so is measured. For fracture toughness tests, the force required to grow a crack intentionally made in the sample is measured.

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The research co-leaders, Professor Robert Ritchie from the University of California, Berkeley and Professor Easo George from the University of Tennessee, first started work on CrCoNi 10 years ago, initially finding the alloy to be exceptionally tough at liquid nitrogen temperatures (~ -200 degrees Celsius). They wanted to take it a step further and see how tough CrCoNi is at liquid helium temperatures (~ -250 °C).

Because otherwise, if the there is only one and the same process capability value to be used after it’s chosen – there is no need the include it to the calculation. Or am I missing something?

Total nominal disk thickness is calculated similarly to done during worst case tolerance stackup analysis by adding the nominal thickness of all disks. Therefore

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In RSS tolerance band is not reduced, you actually allow some rejection compared to worst case tolerance. And the advantage of this allowable rejection can be given to supplier in term of tolerance.

Why do we need to determine the process capability index in the first place? Is the answer that first we need to determine the manufacturing process capability (e.g. 3 sigma) – and second we need to determine the process capability for the assembly “event” (that can differ from 3 sigma)?

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If I calculate the total deviation (not std deviation) I get 1.4832… that is +- 0.74162 (the same as in your calculation). My formula that I used: (0,5+0,5)^2+(0,1+0,1)^2 .. (0,4+0,4)^2)^-0,5 ~ 1.48324.

Informative log Clears my all confusions of tolarrlence stack up analysis… Language used is very easy to understand and satal.

Tolerance Stackup analysis is a design tool used to analyse and optimize product design for assembly. It is used to calculate the cumulative effects of part tolerances in an assembly. Therefore tolerance stackup analysis ensures smooth part assembly on the production floor. In this article, we will discuss the worst case and RSS methods to solve tolerance stack up problems.

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In the above example, the total stacked disks height / thickness will vary from 65.5 mm to 68.5 mm. We suggest you also use this calculator to calculate total disk height using worst case tolerance analysis.

Let’s consider an example of five different size disks with different tolerances stacked on one another. We will calculate the overall maximum and minimum total stacked disk height using Worst Case Tolerance Stackup Analysis method.

A strong resistance to fracture at frigid temperatures make CrCoNi ideal for application at environmental extremes, such as in deep space. Understanding how CrCoNi is so tough brings such uses of the alloy, and HEAs in general, a bit closer.

One question is troubling me. “Standard deviation is calculated by considering manufacturing process is 3 sigma capable.” -> Std deviation = total tolerance/ (2×3). This one I understand.

The alloy, known as CrCoNi, is an example of a high entropy alloy (HEA), which – unlike alloys in use today – contains equal proportions of its constituent elements. HEAs were first developed ~20 years ago, but technologies capable of testing the extreme limits of these materials have only recently become available.

You mentioned in your calculation the direction of the stack. if you have a 6th step, lets say an etching step, that removed a portion of the last disk, would you include the statistical tolerance calculation of that 6th subtractive step into the same way you did for the RSS of the first 5? or would you RSS the 5 additive steps and then get a tolerance analysis done, followed by an RSS of the 5 steps with the subtractive step?

In the above example, the total stacked disk height/thickness will vary from 66.258 mm to 67.74 mm. We suggest you also use this calculator to calculate total disk height using worst case tolerance analysis.

How should we deal with uneven tolerances (e.g. +0.1/-0.3)? Do we “normalize” it to +0.2/-0.2 (& adjust the nominal value accordingly)?

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As you said, Mathematically standard deviation is equal to two times of process capability because the process is distributed on both sides. based on above, for ‘3 sigma’ the total tolerance of each dimension is divide by 6 (i.e. 2×3) to calculate standard deviation. but at the time of total tolerance zone calculation ‘sigma assembly’ multiplied by only 3 for ‘3 sigma’ capacity. my concern is, just like at the time of total tolerance calculation we divide each total dimension tolerance value by 6. like wise we need to multiply by 6 to ‘sigma assembly’ to get total tolerance zone. because process is distributed on both sides of mean value. please provide your feedback.

in above formula denominator (6) defines the distribution of dimensions. Denominator “6” indicates process is 3σ capable with cpk value equal to 1. If your manufacturing process is 6σ capable value ‘6’ need to be changes to 12.

In their most recent research paper, published in Science, Ritchie and George have quantified the toughness of CrCoNi at liquid helium temperatures and used neutron diffraction, electron backscatter diffraction and transmission electron microscopy to reveal how the atomic structure of CrCoNi contributes to its remarkable strength.

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I’ve been told if there is a sequence of just 2 dimensions then RSS is not required. I can’t find where this rule originates. Can you confirm?

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We suggest you read this article on Process Capability Analysis Cp and Cpk for better understanding on Statistical Tolerance Analysis.

Hello, I have the same question asked by Nagaraju. So how does the worst case analysis has disadvantage over RSS method? Compared to Worst case, in RSS, you allow some parts rejection. But what about the cost of those rejected parts? How exactly RSS method is advantageous over worst case?

If you look at the results from Worst case and RSS method. In the RSS method, the tolerance band is reduced. Therefore designers can give more flexibility to manufacturers.

Hi, I like using your stack up calculator spread sheet. Can you share the password so I can name my project? Thanks Sean

Tolerance stackup analysis is used to calculate optimized part tolerances. It can be done manually or using stackup analysis software.

In this example maximum and minimum disk stackup height is calculated by adding nominal dimension with upper and lower tolerance respectively considering required process capability value is 3σ.

Nice writeup. Please note that there is a typo for the 3 sigma area under the curve of the normal distribution bell curve. It should read 99.72% rather than 95.72%. Regards.

Various tolerance stack up analysis methods are used to calculate optimized tolerance values. Each method has its own advantages, limitation, and application. Out of these Worst Case and Statistical Tolerance Analysis methods are more popular. Let’s discuss these methods in detail.

Let’s consider an example of five different size disks with different tolerances stacked on one another. We will calculate the overall maximum and minimum total stacked disk height using RSS Tolerance Stack up Analysis method. RSS tolerance stackup analysis for a given problem can be done in the following steps:

In your bell curve normal distribution curve there is an error – for 6 sigma, instead of 99.72% you have marked 95.72%, and as a newbie I spent 20 minutes to re-affirm that… just wanted to let you know. Thanks for this article and calculator, it’s really helpful!