# Tolerance Stackup Analysis : Worst Case and RSS

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 production floor. In this article we will discuss the worst case and RSS methods to solve tolerance stack up problems.

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

## Why Tolerance Analysis is Required?

Production of closely mating parts without tolerances is economically unfeasible. Tight tolerances can result in 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 part tolerance band. It also helps in reducing part manufacturing cost.

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

## Types of Tolerance Stack up Analysis

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.

## Worst Case Tolerance Analysis

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 extreme limit. For low production volumes, worst case tolerance analysis is recommended.

###### Advantages of Worst Case Stackup Analysis
• Worst Case Tolerance stackup analysis ensures 100% parts assembly on production floor.
• Zero rejection rate.
###### Disadvantage of Worst Case Stackup Analysis
• Worst Case Tolerance Analysis requires very tight individual component tolerances. It increases overall manufacturing and inspection cost.
##### Worst Case Tolerance Stack up Analysis Example

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

Worst case tolerance analysis for a given problem can be done in the following steps:

###### Step-1 Create the Dimension Chain

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.

###### Step-2 Calculate total nominal thickness

Total nominal disk thickness is calculated by adding nominal thickness of all disks. Therefore

Total Nominal Thickness = (15+10+15+12+15) = 67 mm

###### Step-3 : Calculate total tolerance

Next step is the addition of total upper and lower tolerances.

Total Upper Tolerance = (0.5+0.1+0.2+0.3+0.4) = +1.5 mm

Total Lower Tolerance = (0.5+0.1+0.2+0.3+0.4) = -1.5 mm

###### Step-4 Upper and lower limit calculation

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

Max. Height = Upper Limit = Nominal Dimension + Tolerance = 67 + 1.5 = 68.5 mm

Min. Height = Lower Limit = Nominal Dimension – Tolerance = 67 – 1.5 = 65.5 mm

###### Conclusion

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

## Statistical Tolerance Analysis

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 part manufacturing process, machines and other parameters.

Statistical tolerance analysis is based on 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.

Following two types of statistical tolerance methods are widely used in the market.

2. Monte-Carlo Simulation for Tolerance Stackup Analysis

###### Advantages of Statistical Tolerance Analysis
• Statistical Tolerance stackup analysis helps in increasing component tolerance limits. In the worst case, part tolerances become very tight that increases part cost.
###### Disadvantages of Statistical Tolerance Analysis
• When parts are designed using statistical tolerance stack up analysis. Manufactured part dimensions can get out of limits and parts can get rejected. Number of rejected parts depends on part manufacturing capability (3σ, 4σ, 6σ
• Statistical tolerance stackup analysis does not give 100% guarantee for part assembly.
##### Root Sum of Squares (RSS) Tolerance Stackup Analysis Example:

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.

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 following steps:

###### Assumptions

For this calculation we will consider the manufacturing process is  capable and Cpk value is 1.

###### Step-1 Create the Dimension Chain

Similar to worst case tolerance analysis, 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.

###### Step-2 Calculate Total Nominal Thickness

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

Total Nominal Thickness = (15+10+15+12+15) = 67 mm

###### Step-3 Calculate standard deviation for each tolerance

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:

Standard Deviation = Total Tolerance / (2 X 3)

σ1 = (0.5+0.5) / (2X3) = 0.166

σ2 = (0.1+0.1) / (2X3) = 0.033

σ3 = (0.2+0.2) / (2X3) = 0.066

σ4 = (0.3+0.3) / (2X3) = 0.1

σ5 = (0.4+0.4) / (2X3) = 0.133

###### Step-3 Calculate Standard deviation for Assembly

Standard deviation for assembly is equal to root of sum of squares of the individual dimension standard deviation.

σ (assembly) = √ [(σ1)² + (σ2)² + (σ3)² + (σ4)² + (σ5)²]

= √ [(0.166)² + (0.033)² + (0.066)² + (0.1)² + (0.133)²]

σ (assembly) = 0.2472

###### Step-4 Calculate total tolerance zone

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.

Tolerance Zone (For required process Capability = 3σ) = σ (assembly) X 3  =  0.74162

Tolerance Zone (For required process Capability = 6σ) = σ (assembly) X 6  = 1.48324

###### Step-5 : Calculate Upper and lower Limit

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.

Max. Height = Upper Limit = Nominal Dimension + Tolerance = 67 + 0.74162 = 67.74162 mm

Min. Height = Lower Limit = Nominal Dimension – Tolerance = 67 – 0.74162 = 66.2584 mm

###### Conclusion

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

## How worst case and RSS tolerance Analysis Results are Different?

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.

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

## Tolerance Stack-up Calculator

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

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.

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