Wire Ropes Under Tensile Load - Wire Ropes: Tension, Endurance, Reliability - page 125

s

L 2
cos 2 b L 2

DL D ¼ tan 2 b 2 r L 2 x tan b r 2 x 2 L 2 L :

Divided by L, the rope extension by twisting the rope is

¼

p

e D ¼ DL D
L

1 2 r x tan b r 2 x 2

1 : ð 2 : 97 Þ

For a constant rope twisting over the rope length L, the change of the rope
length is

DL D ¼ e D L

and if the twisting over the rope length is not constant, the change of the rope
length is

e D dx ¼ Z

L

dx

p

1 2 r x tan b r 2 x 2

DL D ¼ Z L
x¼0

1 ð 2 : 97a Þ

x¼0

and with x from ( 3.85 ) the change of the rope length is

0
@

1
A dx

s

1 M c 1 dS 0 þ m g x

2

DL D ¼ Z

L

ð Þ 2 r tan b

c 2 d 2 ð S 0 þ m g x Þþ c 3 G d 4

ð Þ M c 1 d S 0 þ m g ð Þ

c 2 d 2 ð S 0 þ m g x Þþ c 3 G d 4

r 2 1

0

ð 2 : 98 Þ

The equation has to be calculated numerically.
The lay angle b 0 of the twisted wire rope is

tan b 0 ¼ u Du
L þ DL D

¼ L tan b L r x
L þ DL D

:

With L þ DL D ¼ L ð 1 þ e D Þ , the lay angle of the twisted wire rope is then

b 0 ¼ arctan tan b r x
1 þ e D

: ð 2 : 99 Þ

and with the elastic rope extension e E the lay angle is

b 00 ¼ arctan tan b r x
1 þ e D þ e E

: ð 2 : 99a Þ

The lay length of the twisted wire rope is