Wire Ropes Under Tensile Load - Wire Ropes: Tension, Endurance, Reliability - page 66

n wj

S ¼ X

n S

X z j cos b j F ij z ij cos a ij : ð 2 : 24 Þ

j¼0 i¼0

According to ( 2.20 ), the wire tensile force in the wire layer i of the strand j is

cos 2 a ij
1 þ m ij sin 2 a ij

F ij ¼ Dl j
l j

E ij A ij ð 2 : 25 Þ

and according to ( 2.19 ) and the wire rope length L = l j cos b j

cos 2 b j
1 þ m j sin 2 b j

Dl j
l j

¼ DL
L

: ð 2 : 26 Þ

Then, using ( 2.25 ) and ( 2.26 ), the tensile force of a wire ij is

cos 2 b j
1 þ m j sin 2 b j

cos 2 a ij
1 þ m ij sin 2 a ij

F ij ¼ DL
L

E ij A ij : ð 2 : 27 Þ

Using ( 2.27 ) and ( 2.24 ), the wire rope tensile force is

!

n wj

X

n S

z j cos 3 b j
1 þ m j sin 2 b j

X z ij cos 3 a ij

1 þ m ij sin 2 a ij

S ¼ DL
L

E ij A ij : ð 2 : 28 Þ

j¼0 i¼0

Combining ( 2.27 ) and ( 2.28 ) by eliminating DL/L, the tensile force in the
certain wire k in the strand l is

cos 2 b l
1 þ m l sin 2 b l

cos 2 a kl
1 þ m kl sin 2 a kl

E kl A kl S

!

F kl ¼ ð 2 : 29 Þ

X cos 3 b j

1 þ m j sin 2 b j

nw

P n S
j¼0

cos 3 a ij
1 þ m ij sin 2 a ij

z j z ij E ij A ij

i¼0

and the tensile stress in that wire is

r tkl ¼ F kl

A kl :

ð 2 : 30 Þ

2.1.4.3 Influence of the Poisson Ratio

The Poisson ratio (transverse contraction ratio) for steel m i = 0.3 can also be used
for the steel wire helix in the strands. Because the length-related radial force
between the wires is very small, the reduction of the wire diameter and winding