The First Reliability Model for Mechanical Situations

A simple reliability model is the stress-load model.
In its simplest form it assumes that the stresses present are normally distributed and that the strength of material is also normally distributed across a number of samples.
With these distributions known, we can calculate the overlap of the two distributions, called interference, and can estimate the reliability of the situation.
The area of overlap is proportional to unreliability. See references [2], [3], [4] and [6] for additional information on this technique.
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Mean of the Load is L = 5000 PSI and θL = 700 PSI L
Mean of the Strength is S = 8000 PSI and θS = 800 PSI
We often desire the Safety Margin to be > 3 to ensure a high reliability.
The Safety Margin unfortunately is a poorly defined term.
You can find at least 2 definitions, depending upon the book. Note – Most books agree that The Margin of Safety is different from the Safety Margin.
The Margin of Safety being the ratio of the average values of the strength and load when the standard deviations are unknown.
The following is the most common convention for the Safety Margin or S.M.
A one time application of a range of stress upon a material will lead to the population reliability of 0.997614 for the range of loads and strengths present.
Figure 1.12 shows the considerable overlap of the two distributions.
Despite this, the one time reliability is still high
This simple model can be employed with a variety of situations.
These are mainly mechanical, but can also be electronic.
Where ever one can describe a probability distribution that is related to a state of a system that has a well defined failure distribution, we can use this approach.
Time may even be added to the whole approach.

Extensions of the simple static model may be based upon the fact that one can model some quasi-static situations by the interference (overlap) of the strength and load distributions.
The distributions represent a probability of strength and a probability of load (stress) in a population of possibilities.
They do not suggest that a single system is changing values of strength, rather the whole population may be drifting.
Load is assumed to be static or drifting just as is strength.
Both distributions are still assumed to be normally distributed.
The math in this case is easy.
The overlap area of the two distributions is proportional to the probability of failure.
This simple model may be extended by adding time or repetitive activities.
The following examples show ways to extend this simple model.
Degradation may be the description of a slowly declining strength distribution.
The stress changing with time may be associated with a number of common failure mechanisms such as loss of lubrication, wear out or damage.

2. Ireson, Grant, Coombs, Clyde and Moss, Richard. Handbook of Reliability Engineering and Management, 2nd. edition, McGraw Hill, New York, 1996
3. Rao, S.S., Reliability Based Design, McGraw Hill, New York, 1992
4. O’Connor, P. D. T., Practical Reliability, 4 th Student edition, Wiley, New York, 2002
6. Carter, A.D.S., Mechanical Reliability and Design, Wiley, New York, 1997

By: by James McLinn CRE, Fellow ASQ

Published in Mechanical Design Reliability Handbook: Simplified Approaches and Techniques ISBN 0277-9633 February 2010 (available as free download for ASQ Reliability Division Members)

Picture © B. Poncelet

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