Wire Ropes Under Bending and Tensile Stresses - Wire Ropes: Tension, Endurance, Reliability - page 262

3.2.9.2 Distribution of Wire Breaks on a Rope

If it is equally probable that all sections of a rope bending length may get the next
wire break, so the theoretical distribution of the number of wire breaks in the
sections of the rope is the Poisson distribution. This is true if the wire breaks occur
accidentally in the sections as the numbers do when throwing dice. The following
conditions have to be fulfilled for the validity of the Poisson distribution, Feyrer
( 1983 ):

• The wire breaks have to occur independently of each other
• The whole bending length should be divided into a lot of sections, l/L [ 10
• The probability that a wire break is found in a unit length should be low, which
means that the wire break rate k = B l /l should be low.
• The reference length L should be greater than the length un it of about Dl = d or
d/2 and the mean number of wire breaks on the sections B L ¼ k L is finite. Of
course, this latter condition is always fulfilled.

Here

B L is the number of wire breaks on the reference length L

B l the number of wire breaks on the fatigue stressed rope length

l (bending length l)

the mean number of wire breaks on the reference length L

B L ¼ B l L = l

B L,max the maximum number of wire breaks on a reference length L

B AL the discarding number of wire breaks on a reference length L

l the whole bending length (fatigue stressed length)

L the reference length

Dl the step length and

z is the number of steps

The probability w of the Poisson distribution that the number of wire

breaksB L = 0, 1, 2, 3, etc. exists on the reference lengths L is

w ¼ B B L

e B L :

L
B L !

ð 3 : 77 Þ

The variance is

V ¼ r 2 ¼ B L :

ð 3 : 78 Þ

The probability that the number of wire breaks is smaller or equal to B L is

p ð B L Þ ¼ X

B B L
L
B L !

B L

e B L : ð 3 : 79 Þ

0