Wire Ropes Under Tensile Load - Wire Ropes: Tension, Endurance, Reliability - page 159

The number of load cycles can be calculated with the methods of the reliability
theory. Without any explanation, Gabriel ( 1979 ) first presented this method in a
diagram for wire ropes of different lengths. The survival probability of the rope
with the length L as a serial grouping of the pieces with the length L 0 is (while
neglecting the influence of the sockets)

P ¼ P L = L 0
0

: ð 2 : 106 Þ

Figure 2.46 shows the rope with sockets and defines the rope lengths. Once
more, the logarithm normal distribution has been used to evaluate Esslinger's
results. In Fig. 2.47 the numbers of load cycles found by Esslinger are introduced
from Fig. 2.45 a nd in addition the lines calculated are drawn for the mean number
of load cycles N and for the number of load cycles N 10 and N 90 , at which point, at
the most 10 % respectively 90 % of the ropes will be broken. The numbers of load
cycles N for the rope length L 0 = 2,030 mm have been taken as the basis for the
calculation because there is only one extreme n u mber of load cycles. For that
distribution, the mean number of load cycles is N 0 ¼ 318 ; 000 and the standard
deviation is lg s = 0.148. The test results and the calculated lines harmonise quite
well.
From the results of ( 2.106 ), an equation can be derived for the load cycles ratio
of the rope lengths L and L 0 [( 2.107 )]. With this, the endurance ( 2.102 ) can be
corrected for different rope lengths. Equation ( 2.102 ) and their constants in
Tables 2.8 , 2.9 and 2.10 are related on a mean rope length of about L 0 = 60d of
the test rope lengths 40d,55d and 100d. Based on this rope length, the numbers of
load cycles—respectively—the rope length factor, Feyrer ( 2011 ), is

1 : 54

f L ¼ N L
N 60

¼

0 : 14 :

ð 2 : 107 Þ

l = d 2 : 5
57 : 5

2 : 54

The results of ( 2.106 ) and ( 2.107 ) depend on the standard deviation of
the number of load cycles. The standard deviation, known un til now for two
Warrington ropes with a t he length 87d is lg s = 0.038 (Fig. 2.40 , N ¼ 82700) and
lg s = 0.092 (Fig. 2.39 , N ¼ 406 ; 0 00) and for the strand 1 9 7 with a length
133d, is lg s = 0.148 (Fig. 2.45 , N ¼ 318 ; 000). The standard deviation is non-
uniform and probably increases as for materials normally found with the number
of load cycles.
As for the rope bending, a mean standard deviation is set at lg s = 0.047 for
the rope length L/d = 60. The standard deviation for fluctuating tension is
probably greater. On the other hand, the rope endurance will possibly at first not
decrease with the rope length as shown in the findings of Suh and Chang ( 2000 ).
With this standard deviation, the decrease of the number of load cycles with the