# Stress Strength Model: Definition

A stress strength model compares the strength and stresses on a system; it is used primarily in reliability engineering but also in economics, psychology and medicine.

In a stress strength model both stresses and strength are considered as separate random variables. Stress experienced by a component is often represented by the random variables designated X; strength of the component is represented by Y. A situation in which X > Y is one in which the stresses are greater than the strengths, and the component fails. If Y > X, the strengths are greater than the stressors.

We can define reliability, then, as the probability a component will not fail: P( X < Y). This, R = P ( X < Y ), is the basic stress strength model, and refining it and applying it to real life analysis is the essence of stress strength analysis.

## Applications of Stress Strength Models

In their landmark book on stress strength models, Kotz et. al details many examples of stress strength models in a survey of scientific literature. These include such applications as:

• Reliability of Rocket Engines: When Y is the strength of a rocket chamber and X stands for the maximal chamber pressure which is generated when a solid propellent is ignited, P( X < Y ) is the probability that the engine will be fired successfully.
• Earthquake Resistance: The strength stress model was used to study the risk an earthquake posed to a particular nuclear generator. With no concrete numbers to define the strength, the researcher took strength estimates from five experts and used the log-normal distribution as a model and a weighted least squares procedure to estimate the strength. A similar procedure was used for the stressor, and the conclusion P(ln X < ln Y) = 0.99978 was reached—a very reassuring number, if accurate
• In a medical study, the reaction of leprosy patients to a medicine was modeled on a P( X < Y ) stress strength model. Initial condition (infiltration status) was taken as X, and Y the change in health after 48 weeks of treatment. The null hypothesis, that initial infiltration values did not affect outcomes, was strongly rejected after an analysis of the data.

## References

Kotz, Lumelskii, and Pensky. The Stress-Strength Model and its Generalizations.
retrieved from https://www.worldscientific.com/worldscibooks/10.1142/5015 on March 11, 2018

Barbiero, Alessandro. “Inference on Reliability of Stress-Strength Models for Poisson Data,” Journal of Quality and Reliability Engineering, vol. 2013, Article ID 530530, 8 pages, 2013. doi:10.1155/2013/530530
Retrieved from https://www.hindawi.com/journals/jqre/2013/530530/ on March 11, 2018

Johnson, R. A. 3 Stress-strength models for reliability. Handbook of Statistics
Volume 7, 1988, Pages 27-54. Retrieved from https://www.sciencedirect.com/science/article/pii/S0169716188070051 on March 11, 2018

------------------------------------------------------------------------------

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

Comments are now closed for this post. Need help or want to post a correction? Please post a comment on our Facebook page and I'll do my best to help!
Stress Strength Model: Definition was last modified: March 17th, 2018 by