# The Factor of Safety as a Statistical Design Variable

This month, we’ll explore the factor of safety as a statistical design variable in a Strength vs. Demand model and how to use a reliability-based service factor to best benefit you and your business.

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### Classic Factors of safety – sometimes called service factors

According to NEMA, a Motor Service Factor (SF) is the percentage of overload the motor can handle for short periods when operating normally within the correct voltage tolerances. It is well known in AC motors that a startup power spike is around 300% of rated load. This type of load is considered a momentary-impact load.
Service factors for application load of motors and gearboxes are based on uncertainty. This could come from any number of individual components or the design process, including, but not limited to, calculations, material strengths, duty cycle, and manufactured quality.  The value of the safety factor is related to the amount of risk one takes in the design process. As engineers and technical people, we never get all the information needed to make risk-free decisions. In addition to the application and design assumptions, there is true variation in every part of the design decision tree. In mechanical engineering, the ultimate strength, yield strength, and fatigue resistance are often used as base parameters in calculation technique. However, these mechanical properties of the material have corresponding bands of variation around the mean of 5% or more and in the realm of metal fatigue it could be even higher. Additionally, the applicability of the calculation theory and its probability to predict reasonable design life can get ambiguous and complicated. In the AGMA PM Gearing Committee, we are currently working on a greatly expanded and refined document for the analysis of bending life in powder metallurgy gearing. In the past, large factors of safety and cautious assumptions have made the prior techniques significantly conservative. More depth and clarity in addition to testing data are resulting in a much greater understanding of the fatigue mechanisms in powder metallurgy materials. (See Figure 1)

### Factor of safety as a statistical design variable

In the midst of all the design and process uncertainty, there is a technique that can determine a statistical- and reliability-based factor of safety analysis. This type of analysis is data-driven, and it is a much more intelligent method to evaluate truth based assumptions. This technique can be used as “Capacity vs. Demand” covariant analysis often used within the theory of constraints. Historically, the technique has been used for call center flow throughput resolution and inventory velocity management. However, in mechanical design, we can use this technique for a “Strength vs. Demand” analysis to determine design and application risk and a corresponding reliability based Factor of Safety.

In the following two examples, we will forgo all of the intensive mathematical proofs to validate the simple results. There are however, a couple of issues to discuss. A statistical service factor can be calculated by assumed probabilities and a chosen level of reliability and confidence. We are not going to do that because we would have to make a series of probability assumptions based on experience, judgment, or educated guesses. Instead, to get closer to truth based assumptions, we will determine statistical reliability based on a limited amount of test data. (See Figure 2)

Consider an assembly line station that runs down bolts in magnesium gear-reducer housings. In a “Strength vs. Demand” analysis, we want to first know what the strength of the bolt or housing threads are. A 100-piece sample of the components is set up to do a torque to failure test. In this scenario, it is advantageous and economical to use surrogate-housing threads of the same cast material and thread engagement so as not to do destructive testing on expensive housings. During the test, only the housing threads stripped. A separate test of bolt strength showed torsional failure about 20% higher than the housing joint. This tells us two things—that the fasteners are robust and that we need only to focus on the housing connection. The following data is taken:

The next test is to determine what the variability is in the “Applied Torque” where our production equipment runs the bolts down to the drawing specification. (See Figure 3)

As a result of taking 100 bolts to failure, we know something about the variation of the threaded joint and by assembling 100 bolts to specification torque we know something about the assembly variation also. The torque gun gives accuracy in this range of +/- 1% or approximately 0.1 Nm in this range.  This gives us our confidence level. Therefore, we can make some statistical inferences about the strength of these components and the process. (See Figure 4)

Determining the reliability results of our actual torque specification relative to the capacity or strength of the joint is as follows:

Where
µSF , σSF = Strength of the joint, Standard deviation of the failure sample
µStressSF =  Applied load to the joint, Standard deviation of the applied sample
Z = The standardized normal probability variable

Since most “Z” tables rarely go beyond (Z = 3.4), we easily conclude that the application and the process are significantly robust to the joint strength. The histograms bear this out:
Since the tails of the Applied and Failure distributions do not cross and based on the samples tested, the process should never result in a torque joint failing. (See Figure 5)

In this next example, a gear tooth impact test was made. It is similar to single tooth fatigue testing. However, instead of cyclic loading, the anvil is dropped from a fixture striking a tooth near its tip at a predetermined height with specific gravitational potential impact energy. (See Figure 6)

Where
BreakData = mean impact energy to crack or shear teeth
ImpactData = impact resistance required by specification

In this case, we can use “Z” to assess the amount of risk there is in damaging gear teeth by application impact loads.

For “Z” = 2.97 there is only a (1-0.9985) = probability of failure of 0.15% and corresponds to a sigma level of nearly 3.0
The histogram bears this out, also. (See Figure 7)

The tails of the distributions cross in the area under the curve to reveal a potential failure rate of 1.5 failures per 1,000 gears in service. This gives us a reliability based service factor that gets closer to the truth with just a short and simple test of tooth impact strength. (See Figure 8)

This technique can be applied to investigations of Capacity vs. Demand or Strength vs. Demand. Normality of the data requirement is not as critical in this type of analysis as it is with some capability studies. However, the technician needs to understand how sample size and variation interact within the process under investigation. Standard deviation is everything. 1.5 failures in 1,000 may be totally unacceptable in many applications. One way to improve that result is to drive down the variation in tooth strength. Another way is to increase the strength of the gear. Either way, use a reliability-based service factor that gets closer to the truth.

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is a senior technical specialist in the development of gearing, drive motors, and power closure devices in the automotive industry. He currently serves on the AGMA Plastic and PM Gearing Committees. Eberle is a master six-sigma black belt and has authored many papers on gearing, measurement system analysis and process statistics. He can be reached at Fred_Eberle@hci.Hi-Lex.com.