Wire Ropes Under Bending and Tensile Stresses - Wire Ropes: Tension, Endurance, Reliability - page 185

or

s

D þ 2 r cos u
D

2

r
tan a 0

dl ¼ þ r 2 du :

The wire length l can be calculated by numerical integration. Then the wire
displacement by bending the strand is

s ¼ l bent ð u Þ l straight ð u Þ

or

s

D þ 2 r cos u
D

2

s ¼ Z

u

r
tan a 0

du r u

þ r 2

sin a 0 :

ð 3 : 3d Þ

0

The maximum wire displacement occurs for u = p/2.
For the half lay length that means u = p, the required length for the wire in the
bent strand is theoretically a little greater than in the straight one. As, however, for
reasons of symmetry no displacement is possible in this position, the wire should
theoretically become a little elongated with a very small theoretical tensile stress

E :

l bent ð u ¼ p Þ
l straight ð u ¼ p Þ 1

r th ¼

For all the wires in the different layers of the strand, nearly the same small
theoretical elongation can be calculated in comparison to the centre wire which is
not elongated. In reality, the great number of layer wires enforces a small
reduction of the centre wire length and a very small elongation of the layer wires
as a small part of the theoretical elongation.
(b) Constant lay angle a

When there is a constant lay angle a of the wire in the strand, there is no wire
displacement in the direction of the wire axis but in the direction of the strand axis
with the angle D # around the sheave centre. Because of the greater space required,
a wire helix with constant lay angle can only occur if the clearance between the
wires is relatively large. Equation ( 3.4 ) is once again valid for the angle # around
the sheave centre as a function of the winding angle u of the wire in the strand

tan u
2

D
2 r 1

2

s

D
2 r

s

D
2 r

# ¼ arctan : ð 3 : 4 Þ

2 2

tan a 1 1