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The time to failure of fiber bundles subjected to random loads

Published online by Cambridge University Press:  01 July 2016

Howard M. Taylor
Howard M. Taylor*
Affiliation:
Cornell University
*
∗Postal address: School of Operations Research and Industrial Engineering, Cornell University, College of Engineering, Upson Hall, Ithaca, N.Y. 14853, U.S.A. Research supported in part by the National Science Foundation under Grant 75-00570 and the Office of Naval Research under Contract N00014-76-C-0790.
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Abstract

The effect on cable reliability of random cyclic loading such as that generated by the wave-induced rocking of ocean vessels deploying these cables is examined. A simple model yielding exact formulas is first explored. In this model, the failure time of a single fiber under a constant load is assumed to be exponentially distributed, and the random loadings are a two-state stationary Markov process. The effect of load on failure time is assumed to follow a power law breakdown rule. In this setting, exact results concerning the distribution of bundle or cable failure time, and especially the mean failure time, are obtained. Where the fluctuations in load are frequent relative to bundle life, such as may occur in long-lived cables, it is shown that randomness in load tends to decrease mean bundle life, but it is suggested that the reduction in mean life often can be restored by modestly reducing the base load on the structure or by modestly increasing the number of elements in the bundle.

In later pages this simple model is extended to cover a broader range of materials and random loadings. Asymptotic distributions and mean failure times are given where fibers follow a Weibull distribution of failure time under constant load, and loads that are general non-negative stationary processes subject only to some mild condition of asymptotic independence. When the power law breakdown exponent is large, the mean time to bundle failure depends heavily on the exact form of the marginal probability distribution for the random load process and cannot be summarized by the first two moments of this distribution alone.

Keywords

FIBER BUNDLECABLEFAILURE TIMEFATIGUELOAD SHARINGRANDOM LOAD
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 3 , September 1979 , pp. 527 - 541
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

†

Dupont trademark.

References

Birnbaum, Z. W. and Saunders, S. C. (1958) A statistical model for life length of materials. J. Amer. Statist. Assoc. 53, 151–159.Google Scholar
Chung, K. L. (1968) A Course in Probability Theory. Harcourt, Brace and World, New York.Google Scholar
Coleman, B. D. (1958) Time dependence of mechanical breakdown in bundles of fibers III. The power law breakdown rule. Trans. Soc. Rheol. 2, 195–218.CrossRefGoogle Scholar
Cramér, H. (1945) Mathematical Methods of Statistics. Princeton University Press, Princeton, N.J.Google Scholar
Harlow, G. D., Smith, R. and Taylor, H. M. (1978) The time to failure of long composite bundles. Technical Report No. 384, Department of Operations Research, Cornell University.Google Scholar
Has'minskii, R. Z. (1966) A limit theorem for the solution of differential equations with random right-hand sides. Theory Prob. Appl. 11, 390–406.Google Scholar
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
Pinsky, M. (1968) Differential equations with a small parameter and the central limit theorem for functions on a finite Markov chain. Z. Wahrscheinlichkeitsth. 9, 101–111.CrossRefGoogle Scholar
Phoenix, S. L. (1978a) Stochastic strength and fatigue of fiber bundles. Internat. J. Fracture 14, 327–344.Google 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
Skorokhod, A. V. (1965) Studies in the Theory of Random Processes. Addison-Wesley, Reading, Mass.Google Scholar
Štratonovič, R. L. (1968) Conditional Markov Processes and Their Application to the Theory of Optimal Control. Elsevier, New York.Google Scholar
Taylor, A. E. (1958) Introduction to Functional Analysis. Wiley, New York.Google Scholar
Vervaat, W. (1972) Functional central limit theorems for processes with positive drift and their inverses. Z. Wahrscheinlichkeitsth. 23, 245–253.Google Scholar