Wire Ropes Under Bending and Tensile Stresses - Wire Ropes: Tension, Endurance, Reliability - page 264

or—for wire ropes which are not too short—it can be simplified by considering the
rope length as a ring and then the number of steps is

z ¼ l

Dl :

ð 3 : 81 Þ

As wire breaks are relatively rare, too small a length should not be chosen for
the step Dl. A realistic length is Dl = d for visible outside wire breaks and Dl = 6
d for magnetic inspection.
The probable maximum number of wire breaks B L,max on the reference length
L is given by

z ð 1 p ð B L ; max 1 ÞÞ 1 [ z ð 1 p ð B L ; max ÞÞ: ð 3 : 82 Þ

For the Poisson distribution is

z ð 1 X

B L ; max 1

B B L
L
B L !

e B L Þ 1 [ z ð 1 X

B L ; max

B B L
L
B L !

e B L Þ: ð 3 : 82a Þ

B L ¼0 B L ¼0

The probable maxim um number of wire breaksB L,max depends only on the mean
number of wire breaks B L and the number of steps z.
Example 3.6 Distribution of the number of wire breaks

Data:
Rope diameter d = 24 mm
Bending length l = 30 m
Total number of wire breaks B l = 150
Step length Dl = 1 d
Reference length L = 30 d

Results:
Mean number of wire breaks B L ¼ B l l ¼ 150 30 24 = 30,000 ¼ 3 : 6
Number of steps z = l/Dl = 30,000/(1 9 24) = 1,250

e B L ¼ e 3 : 6 ¼ 0 : 02732 :

The probability w that B L occurs and the probability p that B L or smaller occurs

Bl w = w = p =

0: 3.6^0/0!*0.02732 = 1/1*0.02732 = 0.02732; 0.02732

1: 3.6^1/1!*0.02732 = 3.6/1* 0.02732 = 0.09837; 0.1257

2: 3.6^2/2!*0.02732 = 12.96/2*0.02732 = 0.1771; 0.3027

3: 3.6^3/3!*0.02732 = 46.66/6*0.02732 = 0.2125; 0.5152

and so on.