The Modelling and Analysis of the Mechanics of Ropes by C.M. Leech

Author:C.M. Leech
Language: eng
Format: epub
Publisher: Springer Netherlands, Dordrecht

and where ε p is the local preslack and x is a probability coordinate (=ε p /ε ps ). The probability of occurrence of a constituent with a preslack, ε p will vary over the number of components or over the structure according to a probability density distribution, p(x). The probability density function is the probability that a specific value of preslack occurs and is thus a weight attached to that value; it is a requirement that the probability density function satisfies the following integral condition

Then or

The above integral assumes that all the constituent fibres are acting; in fact they are all acting when ε ps < ε<ε f , range 2. Below ε ps some fibres are slack (range 1) and above ε f some are broken (range 3); for range 1 the upper integration limit is x = η/λ and the lower limit is 0, for range 2 the upper limit is 1 and the lower limit is 0, and for range 3 the upper limit is 1 and the lower limit is (η − 1)/λ. This applies if λ < 1 or ε ps < ε f that is where there is a small prestrain variability; if there is a large prestrain variability so that ε ps > ε f , implying that some fibres have broken before others have lost their slackness, the upper limit in range 2 is now (η − 1)/λ.

If F b is the maximum achievable structure load (=σ f A) then a variability factor can be defined,

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