Wire Ropes Under Bending and Tensile Stresses - Wire Ropes: Tension, Endurance, Reliability - page 179

For ( 3.2 ), a supplementary relation between the winding angles # and u W has to
be made. Wiek ( 1973 ) and Leider ( 1977 ) presupposed a constant ratio for both of
the winding angles

k ¼ #
u W

¼ h W

2 r W
D tan a : p D ¼

ð 3 : 3 Þ

This means that the lay angle a in the bent strand is not constant. According to
Czitary ( 1951 ), the lay angle is expressed by

1
1 þ 2 r W
D

tan a ¼ tan a 0

cos u W

with a 0 for the lay angle in the straight strand, r W for the wire winding radius and
D for the diameter of the axis in the bent strand.
Schiffner ( 1986 ) found it useful to consider the bending of the strand with the
supposition of a constant lay angle a = const. In this case the length dl of a wire
element in relation to the length dL of the strand element in the bent rope is

dL
cos a :

dl ¼ ð 3 : 3a Þ

The length dL is

d #

D
2 þ r W cos u W

dL ¼ ð 3 : 3b Þ

with the radius D/2 of the strand axis and the winding angle # around the sheave
axis and the winding radius r W and the winding angle u W around the strand axis
(wire helix in the strand). The length dL is therefore
dL ¼ r W du
tan a

:

According to ( 3.3b ), the winding angle d # around the sheave axis is

1

du :

d # ¼ ð 3 : 3c Þ

D
2 r W

tan a þ cos u W

By integration, Schiffner ( 1986 ) calculated the winding angle #

tan u W
2

D
2 r W

1

2

s

D 2
4 r 2 W

s

D 2
4 r 2 W

# ¼ arctan : ð 3 : 4 Þ

tan a 1 1

For a constant ratio of the winding angles, the space curve of the wires must then
be calculated using ( 3.2 ) and ( 3.3 ) and for a constant lay angle ( 3.2 ) and ( 3.4 ) are