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The asymptotic time to failure of a bundle of fibers with random slacks

Published online by Cambridge University Press:  01 July 2016

L. Tierney
L. Tierney*
Affiliation:
Cornell University
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Abstract

Consider a bundle that has been constructed by placing n fibers in parallel between two clamps. The simplest assumption on fiber load sharing is that all non-failed fibers share the load applied to the bundle equally. Phoenix (1978b) shows that, as n tends to ∞, the time to failure of such a bundle is asymptotically normally distributed under certain assumptions on the stochastic nature of the fibers. In this paper we show that this asymptotic normality is preserved but with changed mean and variance if random slacks are introduced, as caused by random variations in the lengths of the fibers. A theorem on random linear combinations of order statistics, which is given in the appendix, is used to prove the result.

Keywords

FIBER BUNDLETIME TO FAILURELOAD SHARINGWEAK CONVERGENCELINEAR COMBINATIONS OF ORDER STATISTICS
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 3 , September 1981 , pp. 548 - 566
Copyright
Copyright © Applied Probability Trust 1981 

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References

Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Billingsley, P. (1971) Weak Convergence Measures: Applications in Probability. SIAM Regional conference Series in Applied Mathematics, Philadelphia.Google Scholar
Coleman, B. D. (1956) Time dependence of mechanical breakdown phenomena. J. Appl. Phys. 27, 862–866.Google Scholar
Coleman, B. D. (1957) A stochastic process model for mechanical breakdown. Trans. Soc. Rheol. 1, 153–168.CrossRefGoogle Scholar
Coleman, B. D. (1958) Statistical and time dependence of mechanical breakdown in fibers. J. Appl. Phys. 29, 968–983.CrossRefGoogle Scholar
Gihman, I. I. and Skorohod, A. V. (1974) The Theory of Stochastic Processes, I. Springer-Verlag, New York.Google Scholar
Neveu, J. (1965) Mathematical Foundations of the Calculus of Probability. Holden-Day, San Francisco.Google Scholar
Phoenix, S. L. (1978a) Stochastic strength and fatigue of fiber bundles. Internat. J. Fracture 14, 327–344.CrossRefGoogle Scholar
Phoenix, S. L. (1978b) The asymptotic time to failure of a mechanical system of parallel members. SIAM J. Appl. Math. 34, 227–246.CrossRefGoogle Scholar
Phoenix, S. L. (1979) The asymptotic distribution for the time to failure of a fiber bundle. Adv. Appl. Prob. 11, 153–187.CrossRefGoogle Scholar
Phoenix, S. L. and Taylor, H. M. (1973) The asymptotic strength distribution of a general fiber bundle. Adv. Appl. Prob. 5, 200–216.CrossRefGoogle Scholar
Shorack, G. R. (1972) Functions of order statistics. Ann. Math. Statist. 43, 412–427.Google Scholar
Tierney, L. (1980) Limit theorems for the Failure Time of Bundles of Fibers under Unequal Load Sharing. , Cornell University.Google Scholar