Safety Factor in Light of the Bienaymé-Markov and Chebychev Inequalities

Abstract

We have amply seen in previous chapters that in order to relate the reliability and the safety factor, one needs the knowledge of the probability densities of the involved random variables. Often, such information is missing. It is shown in this chapter that the celebrated inequalities by Irenée-Jules Bienaymé (1796–1878), Andrei Andreevich Markov (1856–1922) and Pafnutii Lvovich Chebychev (1821–1894) may come to help in finding bounds on the desired probabilities and safety factors. Short biographical notes about Bienayme and Chebychev are given in the Appendix B.

“The Chebychev inequality gives a conservative but quantitative correspondence between safety limit and the lower bound of safety probability. Such inequalities will play an important role even when a certain one-to-one correspondence between safety limit and safety probability is found. Only safety limit will enter the actual design process.”

H.L. Su (1961)

“...The distribution of the statistical variables involved in....analysis (loads, resistance, times of failure, defects, etc.) are neither known nor obtainable by direct statistical inference within the probability range that is significant for safety and reliability analysis.”

A.M. Freudenthal (1975)

“Traditionally, the reliability of engineering systems is achieved through the use of factors or margins of safety...The traditional approach is difficult to quantify and lacks the logical basis for addressing uncertainties; consequently, the level of safety or reliability cannot be assessed quantitatively.”

A.H-S. Ang and W.H. Tang (1984)

“The conventional [deterministic] way of assessing the design is not adequate with respect to the statistical scatter of the design parameters.”

O. Vinogradov (1991)

“We are having fun developing novel insights as well as ingenious, efficient methods of analysis. Let us have more of it. Some insights will be fruitful, and some methods will become powerful every-day tools.”

E. Rosenblueth (1991)