Wire Ropes, Elements and Definitions - Wire Ropes: Tension, Endurance, Reliability - page 49

Equations for round wires. Equations ( 1.9 )or( 1.9a ) and ( 1.9b ) can of course be
used to calculate the cross-section perpendicular to the strand axis of a round lay
wire. With the wire diameter d, the coordinates for the contour of the round wire
cross-section are

a ¼ð d = 2 Þ sin w
b ¼ð d = 2 Þ cos w :

ð 1 : 10 Þ

With these introduced into ( 1.9 ), the coordinates for the contour of the round wire
cross-section perpendicular to the strand axis is

s

r W þ d

2 2

þ d x ¼

2 cos w 2 sin w cos a

0 1

d
2 sin w cos a
r W þ d

d sin w sin 2 a
2 r W cos a

@ A

sin þ arctan

2 cos w

s

r W þ d

ð 1 : 11 Þ

2 2

þ d y ¼

2 cos w 2 sin w cos a

0
@

1
A

d
2 sin w cos a
r W þ d

d sin w sin 2 a
2 r W cos a

cos þ arctan :

2 cos w

The polar coordinates are

d
2 sin w cos a
r W þ d

u ¼ d sin w sin 2 a
2 r W cos a

þ arctan : ð 1 : 11a Þ

2 cos w

and

s

r W þ d

2 2

sin 2 w cos 2 a þ d

2

r ¼

2 cos w

: ð 1 : 11b Þ

The result of the calculations with both of the earlier Eq. ( 1.11 ) is of course the
same as that of ( 1.8 ) but gained in a simpler way with no iteration. Figure 1.40
shows the contour of a round strand wire in the cross-section perpendicular to the
strand axis drawn for r W = 1 and d = 1. In this example, the lay angle has been
given the value a = 60 which is much higher than that used in practice to show
clearly the characteristic difference to the ellipse often used for simplification.